First of all like WoW Skyrim has several Mage Trees so you have to decide what kind of mage you are going to focus on.
Destruction is the standard elementalist that destroys foes with fire/ice/lightning.
Conjuration is the same as WoW's Warlock, you capture souls and use summons to fight along side you. Illusion is your crowd control which is used to manipulate your foes.
Personally I think you should focus on Destruction and Restoration in the beginning. Get Dual Casting and Augment spells early on and you will be extremely powerful. I would only augment Fire first then focus on Restoration. From that tree get Dual Casting and Mana Regeneration.
As a Mage, Mana Regen is critical to your survival. You will cast lots of damage spells and then pause while regening. If done correctly you shouldn't need potions unless there is an emergency. As for survival Heal and Fast Heal are your friend. Have your keys set to quickly change between a dual cast Firebolt and dual cast Heal or Fast Heal.
That being said you will expend alot of mana this way. Thus you should stack only Magic for the first 30 levels. You don't need more than 100 HP because those enemies that can do more can one shot you anyways, thus its a waste to stack HP. Instead plan on once you get hit instantly using Fast Heal to recover 75 HP. As you grow you can add 50 HP can thus it takes 2 Fast Heals (or one dual cast to fully heal)
As I've mentioned above REGEN and MAGICKA are key to survival. That being said your armor should consist of buffs which increase both. Right now at level 40 my mage uses the following:
Archmage Robe: 100% Magic Regen and +50 to magic
Morokei: 100% Magicka Regen
Savos Aren's Amulet: +50 magicka
Ring: +28 Magicka
Hands: +20 Magicka
Feet: +30 Stamina
As you can see I virtually have no armor (only 45 total with most being light armor) The upside is you kill everything very fast. The downside is you heal alot and run to recover magic when necessary and sometimes get 1 shot.
Since then I've been leveling Light Armor and Heavy Armor to 80% and now my gear even in mage robes can take hits with no problem. I can stand there and get hit and barely take anything. Even with 5 enemies on me they barely hurt, then I Fast Heal and instantly are full again. This method plays like a Disc Priest and it virtually impossible to die. Even fighting dragons I can just stand there and heal through it then hit them back.