At his first year at Hogwarts, Ron seems quite proficient in chess, even winning in an impressive game in one of the obstacles to the Philosopher's Stone, which was set by Professor McGonagall.
So who taught him to play?
At his first year at Hogwarts, Ron seems quite proficient in chess, even winning in an impressive game in one of the obstacles to the Philosopher's Stone, which was set by Professor McGonagall.
So who taught him to play?
On J. K. Rowling's official website Pottermore, another wizard's game is described, Gobstones. Just like Wizard Chess is similar to Muggle Chess, Gobstones is similar to Marbles: "Gobstones is an ancient wizarding game that resembles marbles".
The brief description of Gobstones on Pottermore states that
[p]rofessional Gobstone players compete in national leagues and international tournaments, but it remains a minority sport within the wizarding world ... . Gobstones is most popular among very young wizards and witches, but they generally ‘grow out’ of the game, becoming more interested in Quidditch as they grow older. ...
From the statement that "Gobstones is most popular among very young wizards and witches" and the fact that Rowling thinks of it as similar to Marbles we can, I believe, infer that approximately the same percentage of young wizards and witches play Gobstones as the percentage of Muggles who play Marbles: about a handful per class. We can also infer that almost all wizards and witches will know the game and be familiar with the basic game principle, just as most Muggles know of and understand the basics of Marbles.
The description of Gobstones then continues and says that
Gobstones enjoys limited popularity at Hogwarts, ranking low among recreational activities, way behind Quidditch and even Wizarding Chess.
Quidditch is Hogwarts' Soccer, and about as popular as that is among Muggles, so Wizarding Chess lies somewhere between Marbles and Soccer in its popularity among wizards and witches.
If a handful of each school class play Gobstones, and about half of them play Quidditch, then about a quarter of each class should at least know how to play Wizard Chess.
Given such a proliferation, and the fact that sports and gaming are often family activities, we can, I believe, assume that most members of the Weasley family knew how to play Wizard Chess. Ron's grandfather, Percy, and Bill Weasley are explicitly portrayed as playing the game, and when I look at chess players in the real world, it is highly likely that other members of that family played, or at least knew how to play, Wizard Chess, too.
And just as chess players in the read world usually don't learn chess (or soccer or marbles or any other game) from one single person alone, we can, I believe rightfully, assume that with very high probability Ron has learned to play Wizard Chess from first observing and then playing with different members of his family as well as other members of the Wizard society, such as visiting relatives, his childhood friends, and later his class mates.
Because, if there is a Gobstones club at Hogwarts, and Wizard Chess is more popular than Gobstones, is not unlikely that in Hogwarts there is a Wizard Chess club, too.
Ron got his chess set from his grandfather, so it’s possible that his grandfather was also the one who taught him to play. (It’s also possible that he inherited the chess set after his grandfather died, and would then have learned from someone else).
“Ron also started teaching Harry wizard chess. This was exactly like Muggle chess except that the figures were alive, which made it a lot like directing troops in battle. Ron’s set was very old and battered. Like everything else he owned, it had once belonged to someone else in his family – in this case, his grandfather. However, old chessmen weren’t a drawback at all. Ron knew them so well he never had trouble getting them to do what he wanted.”
- Harry Potter and the Philosopher's Stone, Chapter 12 (The Mirror of Erised)
Bill, Ron’s brother, was seen playing chess with him, showing that he knows how to play and was also willing to play with Ron. Bill had already moved out by the time Ron started Hogwarts, but he could have taught him before leaving.
“If Dad hadn’t said anything, old Rita would just have said it was disgraceful that nobody from the Ministry had commented,’ said Bill, who was playing chess with Ron.”
- Harry Potter and the Goblet of Fire, Chapter 10 (Mayhem at the Ministry)
However, Ron may have also learned from someone else who we don’t happen to see playing chess, or he might have even taught himself. If wizarding chess comes with rules on how to play in the box like most Muggle games do, he could have simply read those and gotten good on his own once he knew how to play.
From what we know of Ron’s chess ability, he’d just about have to have learned from either his family or someone he knew before attending Hogwarts, since the chessmen in his grandfather’s chess set knew him well and trusted him when he played with Harry over Christmas break.
“However, old chessmen weren’t a drawback at all. Ron knew them so well he never had trouble getting them to do what he wanted.”
- Harry Potter and the Philosopher's Stone, Chapter 12 (The Mirror of Erised)
This also showed Ron would have been good at chess by then, since the chessmen Harry uses notice his lack of skill and therefore don’t trust him.
“Harry played with chessmen Seamus Finnigan had lent him and they didn’t trust him at all. He wasn’t a very good player yet and they kept shouting different bits of advice at him, which was confusing: ‘Don’t send me there, can’t you see his knight? Send him, we can afford to lose him.”
- Harry Potter and the Philosopher's Stone, Chapter 12 (The Mirror of Erised)
It’s likely he did improve his chess skills when he had classmates to play against (presuming some of the other students were better at it than Harry or Hermione and could therefore provide more of a challenge to him) but it seems unlikely that he got as good as he was at Christmas break during just the approximately four months he’d have been at Hogwarts for.
In addition to what has already been stated, it's also possible he learned against a self-deciding board, like some people in our nowadays society learn from chess computers. As already quoted:
“Ron also started teaching Harry wizard chess. This was exactly like Muggle chess except that the figures were alive, which made it a lot like directing troops in battle. Ron’s set was very old and battered. Like everything else he owned, it had once belonged to someone else in his family – in this case, his grandfather. However, old chessmen weren’t a drawback at all. Ron knew them so well he never had trouble getting them to do what he wanted.”
- Harry Potter and the Philosopher's Stone, Chapter 12 (The Mirror of Erised)
Right afterwards, it's stated that the figures are not only alive, but also capable of giving advice:
“Harry played with chessmen Seamus Finnigan had lent him, and they didn't trust him at all. He wasn't a very good player yet and they kept shouting different bits of advice at him, which was confusing”
- Harry Potter and the Philosopher's Stone, Chapter 12 (The Mirror of Erised)
Which means it's not too far-fetched that the figures could also act at their own will, becoming someone to play against if there's no human opponent. The chess pieces might also actively become teachers, evoking dedicated situations and posing chess puzzles.
There is a part where Hermione says to Ron, "That's totally barbaric." Ron then replies by saying, "That's Wizard Chess." By the way he replied, he obviously had a clear understanding of the game and was very familiar with it. This reveals a lot about Ron's character.
In the book (and movie), the Mirror of Erised shows Ron basking in the glory of being the best Weasley boy. This shows that Ron may feel slightly overshadowed by his older brothers and burdened by the weight of family expectation, and wants to be better.
Therefore, it is most likely that he was taught to play Wizard Chess in his family, and he became so good at it, he was able to battle Professor McGonagall's huge enchanted chessboard that was one of the obstacles protecting the Philosopher's Stone. The fact that he was willing to sacrifice himself shows a lot of maturity at such a young age, perhaps acquired from, as mentioned before, family expectation. (More about that can be found here.)
Because of people expecting certain things from Ron, he might find it difficult for him to feel in control of his own choices -- but he is in control when playing Wizard Chess! This might explain why he is so good at it, and might also explain how he knew to get past the Wizard Chess obstacle:
"It's obvious, isn't it?" said Ron. "We've got to play our way across the room." Behind the white pieces they could see another door. "How?" said Hermione nervously. "I think," said Ron, "we're going to have to be chessmen."
But who in particular taught him to play, I believe that @Bellatrix's answer is most likely correct.
Wizard chess just might have been a regular game in the wizarding families. He might have learnt it like we learn to play cricket and baseball