Quick answer
Draw, decide, discard. The details that decide arguments: your turn begins the moment you touch the wall tile, which also ends your chance to claim the discard, and the claim window closes when the next player racks or discards.
A turn sounds like two steps. In practice it has about six, and the small ones are where table arguments come from: when your turn actually started, whether somebody still had time to call, and whether a tile counted as racked. Here is the sequence, in order, with the moments that matter marked.
The short version
Play passes to the right. On your turn you either take the tile the previous player discarded, if you can use it and claim it in time, or you draw the next tile from the wall. Then you discard one tile, face up, and your turn is over.
You hold thirteen tiles between turns. During your turn you hold fourteen, briefly, which is why every turn ends with a discard: you are putting the count back.
Your turn begins when you touch the wall
This is the detail that settles arguments. Your turn is considered started the moment you touch the next tile in the wall, and from that moment you must take that tile. You do not get to touch it, think better of it, and put it back.
It has a second consequence that is easy to miss: once you have touched the wall tile, you may no longer claim the discard you were still thinking about. Reaching for the wall is itself a decision not to call.
The sanctioned tournament rules state this plainly, and it is worth knowing at any table, because it is the rule that decides who was right when two people move at once.
Pick, pause, rack
Take the tile, look at it, then place it in your rack. The pause between picking and racking is not dithering, it is the window in which an opponent can still claim the discard sitting on the table.
Picking and racking in one fast continuous movement is called out in the tournament rules as unsporting, because it denies the other three players a fair chance to call. Players are asked to leave a beat between the two.
A tile counts as racked when it is placed inside the rack, not balanced on top of it and not sitting in front of it. That definition exists because racking is the moment that closes other people's options.
Name your discard
Say the tile out loud as you discard it. This is part of how the game works rather than a courtesy: the other players are entitled to hear what went out, and a tile named wrongly can be challenged.
A tile counts as discarded once it is named or once it touches the table. Either is enough, which is why a tile you have named is gone even if your hand is still on it.
The claim window, and when it shuts
After you discard, the other players have a window to claim that tile for an exposure or for mahjong. The window stays open until the next player either discards or racks the tile they drew.
It also shuts permanently once the discard is covered by another discarded tile. A covered tile cannot be claimed by anyone, for any set, including for mahjong.
Practically: if you want a tile, say so immediately. Deliberating for five seconds while the next player draws is how a hand gets lost, and the rule gives you no way back.
If you want a joker from somebody's exposure
You may swap the real tile for a joker sitting in an opponent's exposure, but the order matters. Do the ordinary part of your turn first, meaning pick from the wall and rack, or claim a discard and expose, and only then make the exchange.
Exchanging before you have picked is the mistake that gets a hand declared dead. The sequence is not decorative, it is what keeps the tile count honest.
The things you cannot take back
Touching a wall tile, which commits you to it and ends your chance to call. Naming a discard, which sends it whether or not you meant it. Touching a discard you claimed, which commits you to using it in an exposure. And exposing a set, which is public for the rest of the hand.
Everything else on your turn is reversible, which is why the useful habit is to decide before you reach. Look at the discard, decide whether you want it, and only then move your hands.
Put it into practice
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Turn order is the part you stop thinking about after a few sessions. What stays hard is deciding what to discard, every single turn. Mei, our AI coach, reads your actual hand and says which direction it goes for and which tiles are doing nothing for it.