Answer:
You get mini-games by the following:
4-player Mini-games, 2vs.2, 1vs.3:
When a turn ends in Party mode, you play a mini-game.
Battle:
Land on a battle space in Party Mode.
Duel:
Land on the same space as a player in Party mode or challenge them to a duel with the dueling glove item.
Rare:
Dizzy Dinghies - Unlock all other mini-games
Stardust Battle - Complete Story Mode
Mario Puzzle Party Pro - Beat Game Guy's Room (You can go into game guy's room after you beat story mode)
Item:
Land on an item space in Party mode. Sometimes, instead of playing an item mini-game, you will be asked a trivia question instead.
I hope I answered your question!