From what I understand, the way that people get good at chess is by endlessly honing their ability to recognize a wide variety of visual patterns that happen to recur in the game (and especially at the higher levels, to some extent by just memorizing trees of opening moves), rather than by understanding deep principles or learning deep skills that generalize to other domains. As with anything complex that many smart people have spent a long time thinking about, it has a lot of interesting little details in its culture, and as with anything that takes sustained and concentrated effort and punishes mistakes, it probably teaches qualities like persistence and self-control, but on the whole I would definitely recommend against trying to become good at it if you're not having a lot of fun.
That's true, but only of intensely competitive play. You can reliably beat beginners, amateurs, and even many devoted hobbyists by reading about three books worth of visual patterns and then honing your epistemic/instrumental rationality to a fever pitch.
There is a skill level beyond which you cannot win without sinking thousands of hours into memorizing trees of opening moves, but you can comfortably enjoy chess while learning new skills for thousands of hours before you ever reach that point, and even once you reach that point you can always play the occ...
This is a place to talk about hobbies that teach rationality lessons. I'll start with a few:
Programming - Lets you practice math and logic, and gain an intuitive understanding of computation. Also teaches that sometimes you can't argue with reality, but you can fix it. (Also suggested recently by ciphergoth.)
Cryptography - That is, designing/breaking cryptosystems and security protocols. Lets you practice math, logic, and probability theory. Also teaches that almost all human ideas are wrong. After doing this for a while, whenever you encounter a new idea (including your own), you'll instinctively think "If I can't find anything wrong with this, it's probably because I'm not smart or knowledgeable enough or haven't tried hard enough."
Science fiction - Reduces status quo bias and gives interesting insights. Also teaches that the way a society is organized depends a lot on the set of technologies it has access to, so if you don't like how your society works, one lever you have is to change that set.
Video games - Teaches that conventional "success" in life is not much less arbitrary than "winning" in a video game. They're both fine for a diversion, but there are more interesting goals to pursue.
A couple others I've seen suggested in recent comments:
Chess - According to JGWeissman, it "teaches you to carefully consider the consequences of your available actions and choose the action with the best consequences". (I've only played a few games of Chinese Chess, and for me, the lesson was that I don't like competition, and I should look for things to do that nobody else is doing.)
Poker - Lets you practice statistics and overcoming emotional biases.