There are 2 ways to play a Mage.
Either you play a purist one, and you go Alteration and clothes. Dragon hide spell is as good as any armor you can find in game: it reduces the physical damage by 80%, exactly like an armor that would reach the armor cap of 567. The "Pure" mage will be casting destruction at long range mostly. No weapon whatsoever, just Staves.
On a pure mage I also like to use Conjuration, so the melee and aggro is taking care of buy you Atronach or you Dremora Lord. Also use Illusion as Crowd Control with Frenzy and Pacify spells.
A little variant is the Battlemage, who wears Heavy Armor and drops the Alteration tree. I personaly don't like it for roleplay reason. It is not the case in TES, but in most other game Heavy armor, or armor in general, hinder your casting ability and your spells have chances to fail. It is all about roleplay here, so a Battlemage is a valuable way to go for Skyrim
Or you play an "hybrid Mage"
2 Subcategory here:
SpellSword: Mage/Tank wearing heavy armor and using 1h weapon. Destruction for Long range damage and increase the damage of your weapon enchanted with destruction spells. Heavy Armor to be able to sustain melee damage for a long period of time
Nightblade: Mage/Thief/Assassin. magic is use mostly as a defensive purpose. Illusion and Alteration are the main schools used. Damage is done by sneak multiplier, either with bow for long range or with dagger for up-close. Stealth is the key here.
Most of all, it depends on your style of play and how you want to roleplay it.
In Skyrim, if you want to get to the latest level, level 81, you WILL HAVE to level ALL your skills to 100. But it doesn't mean that you have to use them all in your playstyle.