In order to play minigames, you must have either a second local account signed in, or be playing against an online opponent. At the Start Game screen, make sure MULTIPLAYER MINIGAMES is set to "ON." You will then get to play one minigame between each round, for a total of two minigames per game.
There are three minigames in all: Point It Out, Stumpers, and Correct Order. Since you only get two per game, this means that you must play a minimum of two full games to get all three of them at least once each. If you're unlucky, it may take one or two more full games since the minigame you get each time is random.
Note that minigames will NOT be in the game if you're playing alone or against only AI opponents, even if you have the option set to "ON."