What you basically have to do is catch the indicated number of rabbits with the character whose mini-game you want to unlock.

Mario

Sort Or 'Splode: 1 rabbit.

Trampoline Time: 2 rabbits.

Shuffle Shell: 3 rabbits.

Bounce and Trounce: 4 rabbits.

Connect the Characters: 5 rabbits.

Shell Smash: 6 rabbits.

Trampoline Terror: 7 rabbits.


Wario

Coincentration: 1 rabbit.

Bingo Ball: 2 rabbits.

Psyche Out!: 3 rabbits.

Slots Shot: 4 rabbits.

Lakitu Launch: 5 rabbits.

Intense Coincentration: 6 rabbits.

Giant Snowball Slalom: 7 rabbits.


Yoshi

Hide And Boo Seek: 1 rabbit.

Puzzle Panel: 2 rabbits.

Boom Box: 3 rabbits.

Tox Box Shuffle: 4 rabbits.

Which Wiggler?: 5 rabbits.

Unlock Mario Key:
Defeat Goomboss behind 8-Star Door in the Rec-Room

Unlock Luigi Key:
Defeat King Boo in Luigi's Painting on the 3rd Floor of Big Boo's Haunt

Unlock Wario Key:
With Luigi in Mirror Room, hop into Wario's Painting and Defeat the Giant Ice Bully

Unlock White Room Key:
Catch all the Shinning Rabbits

Unlock Cannon:
To access the cannon outside the castle, get 150 stars. There are three 1-ups, a cap and Luigi's rabbit.