I started out with heal and fire, then learned two more in quests.

But now I'm 30 something in restoration and I can't get the trainer at the college to teach me the apprentice heal spell. Over 30 in restoration and still have to use the novice spell I started the game with? If this is right, at what point are you considered an apprentice?

Spells are primarily learned with spell tomes which you can buy from some vendors and court wizards. You can find them around the world and yes some are learned through quests too.

You have to buy the spellbook and then read it.