Another World:
At the "Press Start" screen, press , Right, Left, . If you did this correctly, a icon will appear in the top right hand corner of screen.

Unlock More Stages:
When the Push Start screen shows up, press: , Left, Right, A small icon will show up in the upper right hand corner saying that you have unlocked new stages. The extra stages are in Puzzle mode.

Unlock More Characters:
When the Push Start Screen shows up press: , Right, Left, An icon will show up in the upper left hand corner of the screen saying that you have unlocked new characters.

Thanks to Revolution readers mystic_gohan and teie!