Versus mode is available from the main menu whenever you start the game.
You start with a single versus map unlocked at the beginning of the game and there are 39 in total. You unlock each map by winning a game of versus on the previous one (the first unlocks the second, the second unlocks the third, and so on).
You can play against another player or the AI. The AI difficulty doesn't affect unlocking maps, so if you're finding it difficult, turn the AI down to 1 star to make it a bit easier.