A few pointers...

sneak and pickpocket are your friends.. the first can let you fire off your shot before they know your there the later can gtet you the money you need early on to fund alchemy and enchanting runs as well as getting training in destruction.

Get the fire thingy summons as soon as you can it can tank the lower levels very well. Its very useful for clearing out bandit forts and tanking the leader.

Be SURE to get the best robe you can and a majicka hood... As soon as you can find or enchant them make good enchanted gloves and boots too.

Remember nothing much can outrun you in clothing.. run away.

As soon as you get fireball.. skill 40... go hunting... it realy helps improve your skill. Besides nothing says MAGE like flaming bunnies flying down a mountain.

Do soul runs by buying alot of petty gems and then soul trap wolves and mudcrabs.. this then can be turned into alot of money via daggers and other light weaponry.

As soon as you have money go to the armor shops and look for enchantments you want.. buy em up and disenchant em. Also look for rare weapon enchantments and always go by the general trader stores they often have interesting enchanted items and soul gems.

A few suggestions:

(i) Always put an armour spell up in combat (oakflesh etc). This will help you level alteration, which has some great perks in it. Plus, it's always safer to have the armour, just in case you start taking hits.

(ii) Always have a summon. You can get a flame atronach pretty early, and it's very effective at low levels. Try to keep your summon between you and your enemy: you're a mage, you don't want to be getting into melee.

(iii) If you're using an atronach, use destruction spells of the same element, e.g. ice storm with a frost atronach.

(iv) Seriously consider taking a companion with heavy armour and melee skills as a 'tank', then you can stand behind them and make some use of your restoration spells. Do make sure that you don't confuse your restoration and destruction spells though!

(v) Use 'crowd control' abilities (calm, fear, paralyse, ice form, etc) to neutralise mobs who make it past your defences.

Calm. I think it's a novice spell and will get more powerful with perks and dual casting. You start to get into trouble, make that bad guy your best friend...until you regen some magicka and light him on fire.

Also, Conjuration to let your familiar or flame atronach take the brunt of the damage if you can stay at range.

Don't forget rune spells either, especially in dungeons. Cast them as you advance along narrow passages, and when you hit a big battle in a more open area and need to retreat to restore health or magicka, retreat back along the passage you mined. Your enemies will take damage as they give chase.

The bottom line for playing as a Mage is to stun lock everything to death with Destruction or be your Conjurations sidekick.