Bomber Mansion:
In order to get the Bomber Mansion you must use your bombs to kill yourself or 
die from one of the enemies. The professor will say something about training in 
the Bomber Mansion. Select "No" when asked to continue, and on the main screen 
you will be able to use the Bomber Mansion to train. The Bomber Mansion is a 
training place, and when you complete all five levels you will receive a rank. Your 
score determines your rank. Your score will go down if you lost a heart (you can 
find them hidden in blocks) or if you place a useless bomb (a bomb that does not 
explode or kill anything). 

Planet Bomber world:
Get all ten Lightning Cards from the evil bombers, then defeat the Boss in level 
5-4 (Warhorse). The Planet Bomber world will now be unlocked. In it, you will 
battle two new Bosses (Mujoe and Dr. Mechadoc).

Bonus multi-player battle mode characters: 

Hige Bandit 2: Get all three Red Hige Hige Cards.
Hige Bandit 3: Get all three Blue Hige Hige Cards.
Hige Bandit 4: Get all three Green Hige Hige Cards.
Mujoe: Get all three Yellow Hige Hige Cards.
Mighty: Get all five Jetters Cards.