To unlock Stage Omega you must clear each stage with an A rank or better. To unlock Stage Alpha you must get a C rank or worse. For Stage Omega try to get ALL the 'Quickens' as fast as possible to get a good ranking and also make large combos. To get a bad ranking just dodge the enemies and bullets and then defeat the boss. Get 'Quickens' but no combos.

Unlock Stage Omega:
In Arcade mode get an A rank or better on each stage.

Unlock Stage Alpha:
In Arcade mode get a C rank or worse on each stage.

Complete the following tasks to unlock the corresponding modes in the game.

Unlock Caravan Mode:
Complete Arcade mode at least once to unlock Caravan mode which will allow you to play certain stages.

Unlock Boss Attack Mode:
Complet Arcade mode at least once to unlock Boss Attack mode which will allow you to battle ALL the games bosses.