Yes.
You are give new spells (and abilities) at the Apprentice 1 lvl. Get to your city of choice, find a scribe shop, and you can buy an App 2 upgrade. Rt click, pick scribe option, and now your Static Pulse does more damage (or a heal heals for more, or a buff buffs more - invis might fool slightly higher lvl critters, etc.).
App 3 and 4 are player made. App 4 is a "pristine" combine.
Adepts are next. Adept 1 is monster dropped and rare. Adept 2 so far only comes at "specialized training" points, lvl 10, 20 etc.
Adept 3 is player made, but requires an uber-rare ingredient (and rare monster drop recipe books). Tier one (lvl 1 to 9) uses copper or lapis lazuli. Tier 2 uses silver or coral.
Master 1 is uber uber rare monster dropped.
Note these levels definitely affect your spell or ability's power, but so does your skill (and level, I think). But you do want the best quality upgrades possible, especially for your important and most-used spells/abs.
Summoners want at least adept 1 pets (lvl 10 beetle changes to a wicked spider w/ adept 1).
Hope this helps. GL!