играть в + Accusative is used with games
играть на + Prepositional is used with musical instruments.
Also, играть + Accusative is used to express which character an actor plays.
играть в + prepositional is used for playing on (American English) or in (British English) a team, она играет в футбольной команде = "She plays on a soccer team"
When you say "знаю, как играть (делать, водить (driving) and other verbs) it means that you know how theoretically to do it. If you say "умею играть (делать etc)" - it means that you have had experince in it.
Assuming you mean игру instead of книгу, it's my impression that уметь means being able to, while знать means to know the rules of, while not necessarily being able to play the game. Lots of people know how tennis is play, but not that many people know how to play it.