My first character was a pure mage on Adept, and while the mid levels were difficult due to mana starvation I found that gearing properly with Enchanting can help a lot. I'd advise doing the Mage College early for the Arch-Mage robes and a great headpiece you get on a quest. Also, get a good staff, dual cast adept bolts to stagger and switch to the staff for damage. My advice for a pure mage, using no weapon skills:
1. Conjuration or Illusion will be mostly necessary for crowd control purposes, unless you want neither and would rather roll with a melee companion like Lydia. Personal style or roleplay purposes can dictate this choice. With Conjuration, your summons Atronach summons level with you to a degree [and apparently actually start outdamaging you with their spells at high levels] while Illusion spells are level controlled but can be perked to improve them, so take this into account.
2. Enchanting. Is. A. Must. Start enchanting early and applying Destruction cost reductions to helm, ring, necklace. It's also very easy and quick to level so you'll get a lot of perks early, just perk Enchanting right up the middle of the constellation. I didn't discover enchanting until levels 20+ and if I had it would have made my teen levels far easier. Stacking cost reductions tends to be far, far more effective than either regen % or increased magicka.
3. Pick either Restoration or Alteration. Alteration is good for more passive defenses like armor and magic resistance while Restoration is good for the wards once they're perked for absorbtion and +50% Magicka regeneration, as well as more efficient heals.
4. For Destruction, Impact is a necessity, otherwise, perk it as you will. I'd advise picking two out of the 3 damage elements and focus on those, perking all 3 damage wise tends to be overkill. Runes become obsolete and are magicka intensive so don't perk those.
Alternatively, ignore Destruction and take 1 Hand instead, and be amazed how the game is 10x easier