It's fine for a first game attempt.

– There needs to be a way to restart the game when you die – via a button or at least by pressing a key (e.g. 'R').
– The sprites should definitely use transparency in order to avoid the white boxes around.
– When the player shoots, he probably collides with the bullet which "pushes" him out of the way. He probably shouldn't be colliding with his own bullets in the first place.
– What about sounds or music?

As for health, you'd basically do it in the opposite way to the score. You'd start with "score" 5 and each time you're hit, subtract 1. If you get to "score" 0, the player dies.