To get the Rainbow Run, Dungeon Dash, and Driver's Ed mini-games, you must complete the following tasks:

Dungeon Dash
Buy every 1 vs. 3, 2 vs. 2 and 4 vs. 4 mini-game and go to the mini-game park to get it from the piranha plant chasing the butterfly. Note: This is basically the same thing as Desert Dash from the first Mario Party.

Driver's Ed
Complete the mini-game Coaster under the hard difficulty setting. Go to Woody and buy every mini-game in the game. He will then present with Driver's Ed. Go to Woody to play it at anytime Note: This game is basically a driver's test. This cannot be played during a board game.

Rainbow Run
After unlocking Driver's Ed, go to the mini-game park again. The piranha plant will present with this mini-game and then disappear. Note: This is basically Tightrope Treachery from the first Mario Party.