You start with App1 (apprentice 1). App2 can be purchased from vendors called scribes or trainers in your home city. App2, App3, and App4 upgrades are player made. Check out your town's broker to see if anyone is selling the upgrades you need, or you can even make your own if you're into tradeskills. The level above Apprentice is called Adept. Adept1 spells drop from creatures you kill; people tend to sell the ones they find but can't use themselves. Adept2 upgrades are supposed to be quested, but apparently aren't in the game yet. Adept3 spells are player made from rare harvested items, and usually sell for outrageous prices. I don't know about the 3rd level spells - called Master - since I've never seen one.
Note - you don't have to go in an even progression up the scale from App1 to App2 to etc. You can go right from App1 to Adept3 if you're lucky enough to find the upgrade.