Conjuration, destruction, alteration/illusion, and either restoration, or alchemy. You can throw in enchant as well if you want.
In Conjuration either focus on necromancy, or astromancy (summoning elemental atronachs).
In conjuration focus on 1 to 2 elemental attacks, not all three. Choices up to you which 2 you want.
Alteration is more for defense and situational encounters, while Illusion can easily be OPed in later game due to the obvious AI manipulation Illusion entails.
Restoration for healing, or you can go alchemy.
A pure mage wears no armor, so keep that in mind f you want to stick to a pure mage build. Otherwise you'll be a Battlemage (one who wears armor).