There was a recent article on Wow.com about the difficulty of leveling a mage. While it was focused on what class you can't play, I noticed quite a few people saying that mages were out of reach for them. So I take a moment to offer some friendly advice to the Mage Noob on how to get to level cap.
The first thing you have to realize about mages is that we're squishy. It's a fact of life that if you're solo-leveling a mage, you will die a lot, so don't get butthurt about it and give up. Mages take resilience and a lot of kiting to solo level up to 80. While plate classes can take on elite mobs at equal level or finish group quests by themselves, mages cannot because 2 hits from something like that and you're dead. This does change at level cap with good gear, however. I found myself soloing through a number of group quests in Icecrown the other day. Kiting really is the key to solo-leveling a mage.
There are two types of kiting - single mob and AOE. When you're kiting a single mob, you basically just have to stay ahead of it while keeping it snared and then blasting the crap out of it at a distance. This is possible to do as an Arcane mage at higher levels, once you get Slow. Once you have enough distance from the mob, you can just turn around and start doing your arcane rotation. The key spells to use in a single mob kite while in Arcane are Slow, Frost Nova if the mob gets too close, and Blink to put some distance between you and the mob. There is not magic distance to keep between you and the mob; you just kind of have to find your own comfortable distance. They key is getting back to max range before starting to blast away on the mob again and not letting the mob get within melee range to hit you back. It doesn't hurt to have a Mana Shield up for a little damage, but it's not wise to let the mob beat through it for too long as you'll just wear down your mana pool.
Should you choose to level in Frost, which I recommend, you have a few more tools at your discretion. Frostbolt is going to be your primary damage spell in this spec as a number of the talents in the tree build up it's damage and improve the slow. The nice thing about kiting in Frost is that you don't need a number of spells to maintain the slow and do damage; it's all wrapped up in a nice frostbolty package for you. You've still got Frost Nova and Blink to use in emergencies, and instead of Mana Shield, you have Ice Barrier to absorb extra damage. I recommend keeping this shield up all the time. It has a nice long duration and absorbs damage without draining your mana pool. It also keeps you from suffering pushback while casting should you pick up an extra mob or find yourself being hit by a ranged weapon or spell.
The second kind of kiting is a good way for quick XP and what mages are generally doing when you see purple flashes of Arcane Explosion popping up in the middle of a group of mobs. When I leveled in Vanilla WoW on my priest, I had a mage friend that I used to AOE kite with. He was in Arcane spec when we did this and I'd toss a shield on him to take advantage of the old Incanter's Absorption with a renew so I didn't pull aggro while he blasted away. We used to go through insane packs of mobs like this and the experience and loot was great... until everyone figured out our secret areas and started to do the same themselves.
Now if you're leveling on your own and you don't have a pocket healer, AOE kiting in Arcane isn't recommended. Without something to absorb the extra damage you're taking, you're more likely to die in a swarm of mobs cutting you down while you try to AE them down. Arcane Explosion is not our best AOE spell. So if you're going to try this, I'd recommend doing it in Frost.
Frost gives you the option of slows with Blizzard as well as increased damage with all frost spells, not to mention Ice Barrier and Water Elemental. I kite equal level packs of 7 or 8 mobs in a Frost kite spec at level 80 without taking much damage. If the mobs are close packed enough, all you really have to do is lay into them with Blizzard. I like to keep a Blizzard's range between myself and the mobs, tossing out a Frost Nova when they get just outside melee range to give myself a bit more time till I have to move. If your Ice Barrier is up you can use an AE at close range, but you'll get better damage and more time by just using Blizzard. As for your Water Elemental, taking the glyph isn't going to make him all that useful in this case. Without the Glyph you have a second Frost Nova in Freeze. However if you do have the glyphed version, the extra damage won't hurt either.
If you're going after elite mobs, I'd suggest only taking them one at a time and saving your cooldowns to get big damage in a short amount of time. Some of them are immune to movement impairing effects so you may find yourself having to move more often. It's still possible to kite them with liberal use of Blink, but much more difficult.
If you master kiting, you'll find yourself much better off for leveling and at 80 you'll be able to take quests on you could do on some other classes. Don't forget that it's still a good idea to do dungeons as a lot of good gear comes from the quests involved with them. Once you get up into higher levels you can easily run lower level dungeons for cloth or rep or for your lowbie friends. Once you hit raids you can offer yourself up to the kiting gods on encounters that require it. You may also find yourself enjoying PVP more with this skill in your belt.