Sultan's Game is blessed with having plenty of explanations for what the events that are happening will do. Make sure to read the section on the right of each event (known in game as "Rituals") and to scroll all the way down each time.

You can click on any empty card slot and the game will highlight cards that are compatible with that slot. Hovering with your mouse will also display a tip as to what should be put in.
You can click on any card or character to check its stats and find out more about them.

You can click and drag on the map to move around and you can zoom in and out with the mouse wheel. This is useful to click on some events that sometimes get stuck on top of each other.

"The Grand Game" is one of the first events that are unlocked when you begin the game. This event happens automatically every day regardless if you are present.
This is how it works: every day, a random noble speaks in court. If this person is also a follower of yours, their card will be unavailable that day.
If that person supports you, you'll see their stat contributing on the right side of the event.

At the beginning of the game, most nobles oppose you. It's important to show up so that you can contribute your own support stat and challenge theirs, ensuring it's at least balanced out. If you don't, Slander accumulates and will result in very unforgiving missions to show up.

If the support stat on the right is higher than the "oppose" stat when the event is carried out, you are awarded a random Intelligence card even if didn't send anybody to attend the event. This usually doesn't happen at the beginning of the game because most nobles oppose you for 1 point and you can send in followers to provide 1 support, thus balancing out at zero.

If a person is sent to attend the event, a random Intelligence card is awarded regardless of how it went. This means you can gain up to two if you send a person AND win the support/oppose battle.

If the "oppose" stat wins, the equivalent amount of Slander accumulates. At the start of the game, you can usually only accumulate 1 slander at a time since most characters only oppose you for 1 point but as your relationship with character progresses, some might dislike you a lot and oppose you for more.

If Slander reaches 5, it's removed, your Influence in court falls and a punishing Sultan event is triggered.

What can I put in the item slot next to my character?
If your Influence is less than 5, bringing a single gold coin will award you 1 Influence.
Try experimenting with other items as you see fit.

How do I gain the support of other nobles?
Some support can be gained through special missions certain characters have access to. The others can be met at the Bookstore event and, if your stats are high enough, persuaded into supporting you.

Managing the Estate will produce money.
You want to make sure you are managing it every day to maximize income as gold is a required resource for most events.
Some items can be placed in the item slot. Click on the slot a few times to highlight any potential new items you can put in there.

Even though the tooltip says that "you need a noble to take care of things", that's not actually the case.
You can manage the estate with just 1 non-noble follower by putting them in the left slot. This will yield the same benefits as sending a noble with the same stats.

Managing the estate requires Wisdom and Sociability, the two hardest stats in the game to increase and the rarest that naturally occur in high amounts among followers.
When starting a new game, you can make it a little easier on this front by picking choices that will increase these stats on your protagonist and his wife.
You can employ a "Inside Information" Intelligence card during this event to raise your odds of success.

This event is a good opportunity to gain Intelligence cards. If you score high on the charisma check here, there's a chance to unlock special missions so make sure to bring a character with high charisma.

By default, only nobles can bathe. This makes the event a little harder to do at the start of the game.

Your characters can attend this event to buy a book for 1 gold.
Books can be inserted into Methinks to read them and will buff the stats of the character that reads them.

Each day, a different character shows up at the bookstore.
If you bring a character with enough stats to win the check, you can gain that noble's support in court. More info about this in the section "The Grand Game".

Even if you can't persuade anybody, buying a book if usually worth it for the chance of improving your characters. Some books might even carry secret missions!

Characters, items and more carry stats that influence the events in the game.
Having high stats is important because your stats will be matched against the difficulty of the event that you are carrying out. If the difference is zero or less, you can roll zero dice, meaning automatic failure (you won't be able to use golden dice either).
You need stats that are higher in order to be allowed to roll dice and therefore have a chance to succeed.
There is no downside to having high stats but you want to make sure you aren't putting all your stats onto the same character because there will be many events to take care of around the city and a character can only take care of one at a time.

Wisdom
A very important stat, especially early game as it's needed to do better in "Managing the Estate". very hard to increase, only a few pieces of equipment and some books can do it. Some characters have high natural wisdom.

Sociability
Same as Wisdom except not only is it hard to raise but very few followers come naturally gifted with high sociability. Very few books and equipment can raise it. Extremely important for many routine events.

Charisma
Necessary during most social events. Many pieces of equipment can increase this stat, as well as books.

Magic
Necessary for certain events but it's never used during routine tasks. Books can increase it, as well as equipment.

Stealth
Necessary for certain events. Books and equipment can increase it.

Combat
Necessary for many hostile events. Equipment can increase it, weapons in particular can increase it a lot.

Survival
Necessary for certain events. Somewhat hard to increase but certain pieces of equipment, such as Animals to ride, can increase it a lot. Books can increase it too. Note that only certain characters can ride animals as equipment and your protagonist is not among them at the start of the game.

Each pair of stats is represented on a corresponding Intelligence card that will help you during events (known as Rituals).

It's important to note that some Sultan cards, Intelligence cards and many other things in the game carry stats with them and will contribute their points when used during an event.

Once per day, you can drag your gold coins into Methinks and it will come up with a way to make money for you. If you have a compatible follower, you can potentially earn a lot, otherwise you'll have to wait until the next day. As of right now, there's no reason not to do this as the first thing every day just to see if there's a chance to make more money.

Make sure to also check the section "Managing the Estate".

Attend "The Grand Game" and The Bathhouse! See the corresponding sections.

Some characters will sometimes want to talk to you and ask to be inserted into Methinks. If doing so causes a special event to appear, their card will be unavailable until the event is resolved. Though I haven't tested this thoroughly, you can delay putting them into Methinks if you really need them to do something else that day and they will still be willing to talk to you the following day.

This is also true for certain items that spawn events. Putting them into Methinks and generating the event will "steal" that card from your hand for that day, preventing you from using it into other events.
This isn't a problem most of the time but it's good to know about it once you have learned which cards spawn events when thought about.

Saying anything more than this would spoil the fun!
Besides, like I said in the introduction, this is a rare game in its genre where they bother explaining everything you need if you are just willing to read carefully, as opposed to having to go on a wiki all the time.

So, enjoy it and feel free to experiment, as this game will actually reward you with new and unique paths for your experimentation!