First off, never be afraid to run away. It's what your kind does when faced with a superior enemy.

Most mages stick to one type of Destruction. As a mage, don't be afraid to use ice spells on warriors. They are extremely effective and will serve on annoying those who want to squish you like a bug. Fire is good for nearly everything, shock is really good specifically for mages. Frost for warriors. It will annoy them because they will charge at you slow and have no stamina. Be very afraid when they get close though.

Restoration is useful. It can heal your wounds, make the undead fear you, among other things. I would reach adept in Restoration as well as the 50% Magicka regeneration.

Illusion can be one of the most powerful schools of magic. As Drevis of the College of Winterhold says (join them ASAP) never underestimate the power to confuse your enemies. With Fury or Frenzy, set them against each other. Eventually, you'll control Automatons and undead.

Conjuration is very useful. Having something to back you up will save your life many times. Dremora Lords are generally considered the best summons. The better Conjurers here can elaborate.

Alteration is a school you'll get familiar with. Go for Mage Armor and Magic Resist specifically. That's where your flesh spells come into play and where you'll be able to shrug hostile enemies.

Enchanting is self explanatory. Use it to increase your Magicka, your Magicka Regeneration, or to decrease the cost of spells.

You're gonna want to upgrade your Magicka a lot. Aim for 200-250 Health, whatever in Stamina, and the rest into Magicka. My first Dunmer pure mage started the College at level 15 with 200 Magicka. He left at level 25 with a little more than 500 Magicka.

Having a follower is also a good idea.

Take out those warriors ASAP. If you don't, they'll have a good time.

Also, uh...what's that in the corner of your avatar?

Put most of your points in Magicka. You will need it. Don't chose any dual casting perks until you reach level 60+ in whatever school of magic you are using. By then you should have the magicka you need to manage dual casting. Learn first how to use the spells in each hand, then you will be ready for dual casting. You won't need more than 150 health. A mage stays away from melee range, so you should not be getting hit.

Level up your conjuration and join the College of Winterhold. There you will find the conjuration expert (Phinos Gestor). He will sell Storm Atronachs and Dremora Lords when your skill is at 70. They are expensive both Septims and Magicka wise.

If you meet someone in a tavern wearing black robes and challenging you to a drinking contest, do it. Do his quest for a powerful staff a conjurer will find very, very useful.

Familiar: Once you reach level 5, they're worthless.

Flame Atronach: They are fantastic against Frost Trolls, and lesser Trolls. They offer mediocore ranged support. Do not expect them to tank though.

Frost Atronach: This is your tank. Get Elemental potency at conjuration 80 to unlock his true tank potential. He doesn't deal good damage though... but he's a great iceberg keeping enemies from reaching you. He can also stun-lock a Legendary Dragon to submission if their AI spams their power attack.

Storm Atronach: Ranged support, and the bane of mages, the destroyer of Electromancers. In melee scenarios they are also quite capable, but they excel at range.

Dremora Lord: http://i.imgur.com/q8ZM0Tz.jpg Period.

Seriously though, if you can summon Dremora Lords, then no savage can mess with you. If you have twin souls perk, having two Dremora Lords will make you essentially invincible as long as you stay out of your enemies' reach.

There are also other useful summons if you have Dawnguard and Dragonborn DLC.