To use the Bandit Mini-Games any time go to the stage select screen, hold select and then press L, L, B, A, R. You will be taken to a screen with all six games where you can get unlimited power-ups.
To unlock mini-games to play at your own free will, simply get a score of 700 or more for a world. The bonuses for each world are:
World 1: Flip Card's
World 2: Drawing Lot's
World 3: Scratch n' Match
World 4: Match Cards
World 5: Roulette
World 6: Slot Machine.
Unlock 6 Secret Levels
Complete the game with 100% on all 8 levels to get 6 levels not found in the original Yoshi Island.
You must get 1000 points in a world (100 points on all levels, Extra and Secret included) to make a star appear on the 'Select Game' screen. If you get 5 stars you will be able to play Yoshi's Island on Hard Mode.
You must get 6 stars (100 points on all 60 levels, for 6000 points) then beat the game to see the secret ending.