I've kind of learned the basics of Mahjong, and then the game I ended up playing first is actually not a mahjong game... well kind of. It's Donjara, Bandai's version of Ponjan, which is a simplified version of mahjong. While it wasn't what I thought it was, it was perfect for my skill level!
The simplified rules made it easier to focus on getting a feel for how you're supposed to play in a mahjong game, and since I emulated it, if I messed up I could just reload save states to try out new strategies. In Donjara(note I didn't have a manual and my knowledge on mahjong/donjara are very fresh/incomplete) you only have 9 tiles max and have to get 3 melds of 3 tiles to win a hand. There are no honor tiles(except for something that I think is like a wildcard) and quads as far as I know. It's also much easier as you don't need sets or sequences, you can just have three of the same suit/character. You also have to declare reach once in tenpai, there's no other option, and the game tells you when a tile can put you in tenpai. Because of this, you can't steal your opponent's tile unless you are in reach, so you always have a closed hand. There's no yaku requirement that I'm aware of, but there are technically yaku in that depending on your hand you can get bonus points, in this case usually based on having melds of characters with specific relationships. Depending on the character you play as and their support character, you get different special moves you can trigger before a hand at the cost of some points. The match ends when someone is below 0 points. You play against one opponent, at least in story mode, I don't know if freeplay lets you do 4 players.
As for the Azumanga Daioh skin of the game, it's pretty light. The story is that suddenly a Donjara tournament is being held where all the characters have to travel around the school/town and when they meet each other, they have to play a game of Donjara. Whoever wins 5 matches gets a prize. Each character has their own story where before each match they have a quick conversation. I played through half of the characters stories (6 of 12 playable characters). It's cute, but don't expect anything as funny as the manga/anime.
Spoilers for story stuff
Hopefully the next mahjong game I play is an actual mahjong game lol. Still this one was fun and a good introduction.