I had started this list to help players understand how this game works, and it has since received many contributions from other users. Thank you for this.

Most points here cannot be found as information in the game, while the few points here that are explained in the game are far from clear, such as the artefacts (see [1][2][3][4][5]). Feel free to chip in with more untold knowledge or corrections and I'll update the post.

All information here is now also available in this Steam guide. I hope this list will eventually become redundant as more information gets added to the game itself.

Age transitions (military)

• Siege and naval units are always lost at the end of the Antiquity age. You’ll receive one free cog at the start of the second age once you’ve spent your legacy points.
• Naval units can only be kept at the end of the Exploration age if you have fleet commanders. You'll keep as many naval units as can be assigned to your fleet commanders.
• You'll keep 6 (Antiquity) or 9 (Exploration) of your land units at the end of an age, in addition to the number of units that can be assigned to your army commanders. The only way to easily count how many units you have is by tapping the yield icons on the top of the screen and scrolling all the way down to unit expenses.
• If you have less than 6 (Antiquity) or 9 (Exploration) land units at the end of an age, you will receive the deficit as free infantry units at the start of the new age.
• Should you have more units than can be kept at the end of an age, all excess units will be deleted. The units that remain are upgraded and either assigned to a commander or one of your most populous settlements - though as of yet it's unknown what determines which units are prioritised for deletion, and which units are assigned to commanders or settlements.

Age transitions (other)

• None of your research or study in the current age will matter in the next age. Warehouse buildings and traditions will become available regardless of whether or not you had researched or studied them in the previous age, while all bonuses and other benefits from the tech and civic trees will be lost.
• The settlement limit for each player is at least 3 in the Antiquity age, 8 in the Exploration age, and 16 in the Modern age. Having more settlements than this number will incur a happiness penalty in all of your settlements.
• Although settlement limit bonuses from the tech and civic trees only count for the current age, there are still several ways to increase the settlement limit for the rest of the game. Those are unlocking the two leader attributes found in the Expansionist and Militaristic tree, having selected the Corona Civica memento (unlocked at level 2 with Augustus), and owning the El Escorial wonder. Playing as Xerxes, King of Kings, will also grant you an additional +1 settlement limit per age.
• Buildings that aren’t ageless will now grant +2 (from the antiquity age) or +3 (from the exploration age) of its base yields, and lose their adjacency bonus. While this is generally a debuff and you are nudged to build over them, certain yields will actually be slightly increased this way. For instance, the guildhall will now provide +3 influence per turn instead of its usual +2. Since influence is the scarcest yield, it can be useful to keep all influence buildings from previous ages.
• All civilian units, except for commanders, are lost upon heading into a new age. This includes civilization-unique civilians. Scouts also count as civilian units and are lost.
• Unique abilities of previous civilizations are also lost. Unique improvements and buildings will remain intact, including improvements gained from city states, as they are always ageless.
• Every city except for your capital will become a town. You are given the option to move your capital to one of two different settlements, effectively allowing you to start the age with two cities.
• You’ll retain only a certain amount of gold and influence at the start of a new age. This limit is not very clear at the moment, as it varies between game speeds. You’ll however always gain one free turn of gold and influence equal to the income you have at the start of the first turn of the new age.
• Independent people will always disappear at the end of an age, and you’ll lose any bonuses you gained from city states, including unique resources. Only finished improvements are kept. On the second turn of a new age, a completely new independent people (not yet a city state) will spawn on the location of each independent people lost this way. Having been the suzerain of a city state will mean that the new independent people on that location are neutral to you. Incorporating a city state into your empire is the only way to keep an independent settlement intact.
• You can see the requirements for unlocking future civilizations, as well as a list of unlocked legacy options for the next age, by tapping the lock icon on the top of the screen.
• Mementos can be changed in-between the ages when selecting a new civilization. It's reported that the effects of previous mementos will not be lost this way, effectively allowing you to have six different mementos in a single game.
• Legacy points not spent at the start of a new age are lost. It’s currently not possible to see which legacies you have chosen.

Settling

• Having fresh water (a cyan tile) will give a settlement a permanent +5 happiness bonus. Navigable rivers grant fresh water to adjacent tiles, while non-navigable rivers only grant fresh water when settled on. Several other tiles, such as oases, will also grant fresh water.
• Exceeding the settlement limit will give each settlement a -5 happiness penalty, down to -35. Settlements with negative happiness will lose -2% of their yields for every negative happiness point.
• Settlers can be trained in any settlement that has at least five population, and will not consume any population.
• Using a settlement to claim a tile that has a "goody hut" on it will not grant you any benefits, unlike in previous Civilization games. You must walk onto the tile with any unit or raid the tile with a naval unit to trigger the narrative event.

Combat

• Naval units can attack districts and land units at range, but are forced to engage in melee combat when they attack an embarked unit or another naval unit.
• War support does not grant you any benefits, but instead penalises the opponent. Per negative point, they lose -1 strength on all units and a static amount of happiness in all of their settlements. The happiness penalty is -3 per negative point in settlements they have founded themselves, -5 in settlements founded by someone they're not at war with, and -7 in settlements founded by you.
• You must first gain control of every fortified district in a settlement before it can be conquered. Note that the Dur-Sharrukin wonder also counts as a fortified district, but does not show any walls. Conquered or traded cities will become towns until upgraded again, which cannot be done until the unrest in the settlement passes over.
• Conquering a settlement with a wonder will reportedly give you all the benefits of that wonder as if you've built it. For instance, a settlement with the Terracotta Army will grant you a free army commander. Regardless, conquered wonders do not count towards the cultural legacy path of the first age.
• When razing a settlement, you're warned that this will give all your current and future opponents a +1 bonus to their war support. This however only lasts until the end of the current age.
• Due to an oversight, units heal more health from pillaging tiles at faster game speeds than what is shown, as the displayed number is meant for the standard game speed. On the other hand, less health is gained at slower game speeds.
• Having a military unit on a tile of a settlement of someone you are at war with will prevent that player from constructing anything on it, and any on-going construction on that tile is halted. Players can also not select tiles occupied by an opposing military unit when a settlement expands.

Commanders

• Commander skills do not stack. Try investing points in different skills if you're fighting with multiple commanders. Experience from unit actions is always equally shared between all commanders in range.
• Commanders can’t outright die - they will respawn in the capital after several turns when killed, retaining their promotions and experience. The amount of turns is not quite clear, and might vary per game speed.
• You can assign either a single settler or scout to each army commander, as long as there's still a slot available. Commanders themselves also have the "add to army" button, due to an oversight, but this cannot be used.
• Army commanders can have six units assigned to them once they've unlocked the Regiments skill in their Logistics tree.
• Units unpacked from a commander will have no movement points left unless the commander has the Initiative skill in their Assault tree. With that skill, land units can even be unpacked in water tiles without their usual movement cost for embarking.
• Outside of war, commanders can be placed on any city hall or palace to reduce unhappiness in that settlement by 10%, plus another 10% for each promotion.
• There are three different types of aircraft commanders - aerodrome commanders, squadron commanders, and aircraft carriers.
• You'll receive an aerodrome commander in each aerodrome you build, and this is the only way to get them as they are assigned to that aerodrome and cannot be moved.
• Squadron commanders can be trained in all settlements with an aerodrome. They can be moved between aerodromes and temporary airbases, which they can build themselves. Airbases can be built on any flat tile within eight tiles of a settlement you control.
• Aircraft carriers, which are in fact commanders, can be trained in any city with a naval district. Since aircraft can only travel up to a certain distance between aerodromes and airbases, aircraft carriers allow you to move them greater distances. Additionally, all aircraft can be directly rebased onto any aircraft carrier in range, including aircraft carriers there were constructed on the current turn.

Movement

• Moving over flat terrain or any tile with a road will not affect a unit’s movement. Without a road, all rough terrain, non-navigable rivers, and terrain with trees (woodland, rainforest, taiga, or steppe) will deplete all of a unit’s movement, regardless of how many movement points it had left. 
• Not all districts have a road, which is simply strange and inexplicable, and means you'll have to hover over a district tile to see in its tooltip if it has a road. The district with a city hall will always have one.
• Naval and embarked units can move over navigable rivers and coast tiles without their movement being affected, in addition to ocean tiles once Shipbuilding is researched. Embarking or disembarking will always deplete the unit’s movement, unless the unit is in range of an army commander with the Amphibious skill in their Maneuver tree.
• When a unit enters an ocean tile before Shipbuilding is researched, its movement is depleted and it takes any number of damage between 11 and 20. AI takes slightly less damage from this.
• Moving a unit onto a bridge built over a navigable river will remove its cost of embarking, although moving off the bridge will still deplete the unit’s movement. Bridges built in previous ages lose this strange benefit.
• Scouts are an exception to most movement rules, including embarking and disembarking. Their movement is not affected by anything else than non-navigable river tiles.

Policies and diplomacy

• The number of turns remaining until your next celebration is shown in the overview tab of the social policies menu. When you trigger a celebration, any excess happiness is saved up for the next celebration. If a new celebration would happen while you are already in one, it occurs immediately after the current one ends.
• Some civilizations gain bonuses for the use of traditions. These are the only policy cards that remain available between ages and have a noticeable feather icon in the policy menu. Traditions are unique to each civilization and are found in their own civic trees. Once again, traditions not studied in a previous age will still be unlocked.
• Ideologies are chosen in the third age, also in their own unique civic trees. You may only unlock a single ideology of the three given options, and this cannot be changed later. Although each ideology has different benefits, it’s entirely possible to finish the age without ever choosing one, and this may in fact save you from neighbours who would’ve become angry at you for your ideological differences.
• Though you can accept any incoming requests to start an endeavour, certain endeavours can only be requested if they are related to your leader. For instance, you can only request the Research Collaboration endeavour if your leader labelled as Scientific (as seen when selecting your leader at game creation).
• While espionage actions have a strong impact on the game, they’ll also negatively affect your influence. If your espionage action is revealed, your influence per turn will drop for a while. If you are spying someone while they are counter-spying against you, your influence per turn will also greatly decrease, as the cost for finishing the espionage action against them will increase. Exact numbers are unknown.

Claimed tiles and improvements

• Worked tiles not improved by districts are considered rural tiles. Each rural tile equals one rural population, and each building or specialist equals one urban population.
• Unique improvements, such as the Great Wall or Terrace Farm, as well as those from city states, can be built on rural tiles too boost the yields. In short, these improvements will keep all current and future yields of the tile (minus one food or production). For instance, if you replace a farm with a unique improvement and later build a granary, the tile will still be given +1 food.
• Building a unique improvement on a tile that already has one will remove all bonuses of the former improvement.
• Each settlement can only claim a radius of up to three tiles from its centre. There's currently no way to swap tiles between settlements.
• If a settlement has no available tiles or districts to work on when it grows, a migrant will appear in the settlement. This migrant can be sent to another settlement to improve an unworked tile.
• Natural wonders provide its bonuses to each settlement that owns at least one of its tiles - not just the first settlement.
• The natural happiness of a tile is related to its hidden appeal, which is in some way affected by whatever is on the adjacent tiles. Floods and other natural disasters may also affect yields, but how exactly any natural yields are determined remains a complete mystery.

Buildings

• The palace building in the capital gains a +1 science and +1 culture adjacency bonus for each adjacent "quarter", which is any district with two buildings.
• Generally, food and gold buildings receive an adjacency bonus from navigable rivers and water tiles, production and science buildings from resources, and culture and happiness buildings from mountains and natural wonders.
• Constructed wonders grant adjacency bonuses to all buildings except for warehouse buildings, the city hall, and the palace.
• Without modifiers, each specialist costs -2 food and -2 happiness to maintain, and grants +2 science, +2 culture, and +50% to the adjacency bonus of the buildings in the assigned district.
• Buildings will usually cost -2/-3/-4 happiness and -2/-3/-4 gold to maintain. Happiness and gold cost increases by one for each age, based on when they were built. Happiness buildings do not have a happiness penalty, and gold buildings have no gold penalty. Warehouse buildings have no maintenance costs at all, but also have no adjacency bonuses.
• Buildings can be placed next to a finished wonder as if they were a district, as long as the wonder is adjacent to another district in the settlement.
• When within the settlement details menu (the list icon), all districts and improved tiles will have a coloured outline. In case you forgot where you placed something, you can hover over a building in the list to highlight the tile where it's built.
• Population lost due to damage will return when an affected tile or building is repaired.

Trade

• You may only trade with foreign settlements that have at least one worked resource, unlike in Civilization VI. Treasure fleet resources in the second age do not count as they cannot be traded.
• Effects of all resources stack additively. Having five silver, for instance, will grant you a +100% gold bonus to purchasing units, effectively cutting the cost in half.
• Resources can only be assigned to and from cities in range of your trading network. Building any naval building in a settlement will usually add the settlement to the trading network. Trading range may also be increased with a town specialised as “Trade outpost”, or by having a merchant manually connect two of your settlements. It's not clearly indicated at all why a settlement may not be connected, so you just have to try these things.
• Resources cannot be reallocated in-between turns until a new resource is obtained, or the amount of resource slots in any of your settlements increased for whatever reason, such as by building a market or by slotting a certain policy card.
• Towns turn all of their production into gold. Towns that are not set to “Growing town” will additionally provide all of its food to each city in its range, causing the town itself to stop growing. This range appears to be shorter than the trading network range, but it’s not known how short. As of yet, you can only use the town details (the list icon visible when you select a town) to see which of your cities the food is sent to. If there are no cities shown to be in range, the town continues to support itself.

Religion

• Your missionaries will only be able to spread your own religion, even if they were created in a settlement that follows another religion.
• Holy cities cannot be converted to another religion, not even after the founding civilization is completely erased from history. It's therefore not recommended to take the Brahmanism belief (relics for conversion of capitals).
• Independent people cannot be converted to a religion until they become a city state.
• The second and third founder beliefs of a religion can only be unlocked via very rare random events. It’s completely up to chance whether you’ll ever see these.
• Both the urban and rural population of a settlement must be converted to fully convert that settlement, as explained in the legacy path. If the two populations follow a different religion, the rural symbol is coloured red. However, due to a bug, the red colour unintentionally remains even after both populations follow the same belief. Reloading will fix this confusing issue.
• There’s currently no way to know the share of rural or urban population of a settlement other than counting every tile it has and hoping you got it right. This is very detrimental for the Lay Followers and Ecclesiasticism beliefs (relics for settlements with at least ten rural or urban population).

Treasure fleets

• Once you’ve researched Shipbuilding, settlements in distant lands can produce treasure fleets. These settlements require a fishing quay and must be working on any resource that mentions treasure fleets in its tooltip, such as sugar or tea. You'll also need a fishing quay in your capital or any other settlement on the home continent connected to the capital.
• You can see how many turns it takes to produce the next treasure fleet in the resource menu or in the details of a settlement (the list icon).
• Treasure fleets can be emptied within the borders of any of your settlements on your home continent, providing points on the economic legacy path equal to the amount of treasure fleet resources that the original settlement was working on.

Factories

• Factory resources can only be worked in settlements with a factory. Both the resources (unless imported) and the selected settlement must be connected to your capital via a port or railroad, and the capital itself must also have a port or railroad.
• Factory resources are empire-wide, and you'll receive one economic legacy point per turn for each factory resource slotted to a settlement. You can only slot one type of factory resource to each settlement, because you are meant to slot multiple copies of the same resource to a settlement.

Artefacts

• Selecting an explorer will show an overlay of all known artefact spots (the shovel icons). Explorers can be sent to any museum or university (including foreign ones) to discover all yet undiscovered artefact spots on the same continent as that building. These buildings are highlighted with a vase icon. Note that the university can no longer be built in the Modern age.
• Initially, only artefact spots from the Exploration age are shown. You must study the Hegemony civic before explorers can also discover artefacts spots from the Antiquity age in a museum or university. However, as soon as any player has revealed the artefacts on a continent, they become visible to all players. Even players without the Hegemony civic can dig up Antiquity artefacts once someone has discovered them.
• Each civilization digging at an artefact spot will receive one artefact when the digging is done. There does not seem to be a use for sending more than one explorer to the same spot, even though the AI keeps doing so.
• Artefacts are also randomly found when overbuilding.

Force-ending turns (PC-only)

• Force-ending a turn is a PC-only mechanic that has also appeared in the previous games, and can be done with Shift + Enter.
• This mechanic is frowned upon in multiplayer due to its exploitable nature. It allows you to skip everything that’s left to do on your turn, while saving up all your unspent research, culture, and production. For instance, if the civic for a wonder takes three more turns to be studied, you could use this mechanic to save up the production of a certain city for three turns, thereby saving three turns on building the wonder in that city once it can be built.
• Yields saved this way are only lost on age transition.
• Force-ending turns can also delay celebrations and several other choice events, including having to support an ally that goes to war. However, you can't avert crises this way, as a crisis policy slot will automatically be slotted in for you if you try.

Some more useful things to know

• "Legend unlocks" seen in the leader attribute trees can only be selected once you reach a certain level with a leader by playing enough games with them. Reaching a higher level with a leader may also unlock more mementos and legacy options selectable at the start of an age. Leader progress and unlockables can be found at game creation or in the main menu.
• On PC, the cutscenes at the end of an age can be skipped with the Esc button, and you can select the "Show more" button in the pause menu during a game to quickly exit to desktop.
• Also on PC, you are able to recover autosaves lost during an age transition from a backup folder (located under ~\Documents\My Games\Sid Meier's Civilization VII\Saves\Single\auto\prev). Moving the files out of that folder into the auto folder will show them again in the game.

Several common bugs you should know

• Not being able to claim a tile that was previously owned by a (now-destroyed) city state. This has no fix as of yet, and may prevent you from expanding a settlement.
• Not being able to generate treasure fleets in a settlement that meets all the requirements. I was told this issue is related to the fractal or shuffle map, and has no known fix.
• The Dogo Onsen wonder should not grant every settlement in your empire +1 population on a celebration. It’s a fun broken thing, but it also breaks the late game growth.
• Army commanders with the Merit commendation (+1 command radius) will still only receive experience from adjacent units.
• Not being able to build wonders when all requirements are met. This is seemingly caused by cancelling a building that was already in the queue on its first turn, and this can only be resolved by completing that building or entering the next age.