The School of Magic
The player may find it lucrative to take their robes to the College of Winterhold as early as possible. While not being insurmountable, should you aim to fulfill the College's Quests at higher levels without significant practice in your magic disciplines of choice, the enemies you'll face might not be easily or readily vanquished by employing your mystical means only.
Upon arrival at the College, you will need to pass a test given by Faralda. She will require you to demonstrate proficiency with magic before granting entry. If needed, she will sell you a spell for 30 gold. Faralda will also offer you training in the Destruction skill, as well as sell you additional Destruction Spell Tomes.
Other senior members of the College, each specializing in a School of Magic, will also offer you training and sell you Spell Tomes. Seek Faralda for training in Destruction, Phinis Gestor for Conjuration, Tolfdir (the instructor for your first class) for Alteration, Drevis Neloren for Illusion, and Colette Marence for Restoration.
You can buy spells up to the Expert Level. Once you near the Master level, you can start a quest in each school of magic to earn the Master Magic Spell.
See: Master Magic Quests
You will later gain one of the more powerful and rare robes in the game after defeating a certain quest at the college. This robe is helpful for those wanting to enhance more than one of their spell types (ex: Destruction, etc).
The Mage Stone, which increases the learning speed of magical talents by 20%, is very helpful for this quest line.
How to Learn Spells
Spells in Skyrim are learned by reading Spell Tomes. Selecting a tome from the player's inventory adds the spell to the Magic menu and destroys the tome. Spells are classified by levels (Novice, Apprentice, etc) but may be cast by the player regardless of skill provided the player has sufficient magicka.
Two aspects to reinforcing magic include fortifying magicka (increasing the pool of magicka a player has) and fortifying the regeneration rate of magicka. It is important to note that magicka regeneration is severely lower during combat; thus, increasing the amount of magicka could be a more useful path.
Spell Tomes stocked by vendors and randomly generated as loot from dungeons will be appropriate to the player's skill level (for example, Adept level spells become available when the player's skill reaches 45-50). Most vendors' stock of tomes seems randomly selected, though vendors at the College of Winterhold will typically offer the full selection of spells up to the player's current skill level.
Early in the game, upon reaching Riverwood, seek the merchant Lucan Valeris. He sells basic Illusion, Destruction, Alteration, and Conjuration Spell Tomes.
