You could unlock a spaceship in UPGRADE.

After that, you could upgrade the spaceship by tapping the button below. BULLET LIMIT or POWER of your spaceship will be enhanced automatically after the consuming of coins/crystals/ship pieces. 

BULLET LIMIT means the upper limit level of your spaceship in a stage. The level of the spaceship could be increased by getting Booster-Power Up or Booster-Power Max. When it has reached the upper limit, getting more Booster-Power Up will not increase the strength of the spaceship anymore. 

POWER means the attack ability of the spaceship.