Punk:
Shoot the powerup carrier to wake it up, then let it run off screen, it will return carrying something, fairy, large gem, or bomb powerup. If you destroy it before it gets off the screen you get nothing. They are all preset, no way to change what they bring. Fairy carriers always bring fairys.
Experience is how you level up, everything you do gives experience, but the orange gems give the highest boost. Personally though, I don't worry about the colors, just collect them all and not die and you should hit max level by stage 3. This game has no rank afaik. So yeah you want max power all the time.
The diamonds next to the experience bar is your bomb-power level indicator.
As for beating the game, the best strategy is just put shot and bomb on full auto. Then it's just memorization really, know where to be and the first loop is quite doable. I can clear both loops without much effort these days. For spells, I use them in trick spots when I want some invincibility, but the shield and fire-fairies worthless compared to the good old full-auto spam strategy.