I know that when you get to level 9 you get a gem powerup or something, but what happens at other levels? If you get past level 9 do you get a prize or something?
Nothing really happens. As you mentioned, you unlock the gem ability at level 9, but that's about it. They might add something in the future.
You do however get +1 score multiplier for each level-up objective you complete. So if you're going for that Radical High Score™, you're gonna want to complete all of them.