I'm looking for a note-by-note music making program where I can make my own rhythms, arrangements and melodies (just for fun, nothing serious) that isn't super expensive. So something like Mario Paint, but better than Mario Paint. Do you have any ideas?
This is what you're looking for. It's the best you're going to get for free.
If you're going to want professional level software, you're going to want Finale: http://www.finalemusic.com/ or Sibelius: http://www.sibelius.com/home/index_flash.html
You're going to have to pay Photoshop level prices for either, though. Or you could pirate them like the asshole you probably are.
Musecore should be just about everything you need if you're "just for fun", though.