Power-training spells and where to get them

Mysticism: Minor Life Detection. Dude in Mystic Emporium, Imperial City Merchant Quarter.
Alteration: Open Very Easy Lock (trust me). Same dude from Mystic Emporium.
Conjuration: Bound Dagger. Alberic Litte, Chorrol Mages Guild.
Illusion: Inspiring Touch (trust). Delphine Jend, Bravil Mages Guild.
Restoration: Minor Heal. You start with it whether Res is in your major skills or not.
Destruction: Flare.

Destruction is hard to power level early on, because you need viable targets for all of the above spells, and Des doesn't have any self-targeting spells before you get access to a spellmaking altar.

I said trust me for Open VE Lock and Rally. Those two spells are so cheap that if you're playing a non-atronach and you didn't spastic up your character build by eg. playing a redguard then you should regen your mana before you are able to cast it again. Unless you rapid-cast, which you should do anyway at the start to kick things off. If you are an atronach, well do it by Lady Luck at night and the Dark Brotherhood listen will be there to Rally and you can target a door on the second level facing roughly in the direction of the mages guild for Open Lock.

Minor Life Detection, Minor Heal Wounds and Bound Dagger are classic textbook power-training spells. Bound Dagger has the trick:
1. unarm yourself (if applicable)
2. cast spell
3. stow weapon (dispells)
4. go to 2

The last spell school is Alchemy. This should be obvious: make potions. Spam them out all over the place and sell the results for quickcash TM. Pick things as you walk past them, raid your guildhalls, go into peoples homes when they are asleep and raid the cupboards, barrels, boxes, tables and baskets for foodstuffs and pump out restore fatigue potions.