I am in the midst of my first as well. Regardless of what you choose as your specialty (I use a lot of fire & shock most of the time and only occasionally frost), but find that both restoration and alteration are invaluable. I don't really use a lot of conjuration, but use the following method to 1st get the novice perk and then the apprentice asap and then you can decide where you want to go from there. They save your butt all the time and you'll need the perk for Alteration, conjuration and restoration as well as illusion. You'll need the flesh spells (unless you're going to wear armor - which is an option since 50% regen with reduction in either conjuration or destruction is available on vampire armor by about level 4). You don't really have to worry about destruction too much since it will level quickly and flames, frost and sparks are perfectly viable well in to the game. Note on destruction: Don't bother with the dual cast perk right away. I find it much more useful to just cast on both hands (especially with an Altmer). It does double damage and, once you get the initial perk, it doesn't cost that much and can take out just about anything.
I suggest Altmer if you're not going to concentrate on destruction, although Breton is great too if you're in to conjuration (you already have a spell to start with that can be very useful) and the resistance can be a great asset, although you won't run in to a whole lot of spell casters early game.
1) Use you spells in Helgen, but stay back and let the brutes do the fighting. Fury and flames are useful.
2) Get your mage stone right off (you can always switch later, but it helps early on).
3) Be sure to purchase your frost, raise zombie and oakflesh spells (fury if you're playing anything other than Altmer as well).
4) Methodically make your way through every encounter (including random) at all locations between Riverwood and Falkreath (they do respawn between the two locations regularly).
5) At every encounter cast; Raise Zombie, Oakflesh and then use your preferred destruction spell to destroy your creation. This will raise all those very quickly (particularly conjuration - I've hit 50 before level 6). Watch what you raise though - bandits and worshippers aren't usually an issue, but Thalmor and some mages can give you trouble.
6) If you're not in to your 50% robes, you can either hit Falkreath Watchtower and get 75% Necromancer robes or just kill some Vigilan (again - somebody to be careful of when you raise their corpses - they can be tough) if you want boosts to Restoration or Alteration. Alternatively - if you don't find any vampires right away, hit Shriekwind - you can usually get some there or at least some Adept robes before level 10.
7) Once you've build up about 2-3K in gold, head right away to Whiterun (you don't have to actually do any quests - just get there!) and get your adept level spells from the court mage. The bolt spells are invaluable if you want ranged in the upcoming encounters. Be sure to get Fast Heal (Healing Hands too if you have a follower), Stone Flesh (it's costly, but, with the perk it's not so bad and it will raise your Alteration), the next level raise dead spell (I forget if it's Raise Corpse or Revenant), if you're in to bound weapons you can get the sword now too or you can get it in Falkreath as well, Conjure Fire Antronach (hey - it's a good ally and will raise your Conjuration even further) and most definitely Muffle! In re to Muffle, use it all the time! It will raise your Illusion very, very quickly with use. Be sure to get the Illusion perks as well up the right hand side until you get Silent Cast. That one is awesome!
Now that you've done all of that, if you want to do some quests or exploring or adventuring - go for it! Me, I do the Dragon Rising thing and get Breezehome so I have a base and and alchemy table (I find alchemy, while I don't use it very much for actual potions - I use it for cash). On that note - be sure to collect the ingredients for Elswyr Fondue - you'll come to love it and, if you want a good early potion for cash do moth wings and vampire dust - not only can you regenerate with it, but you can also go invisible for a good period.
In any case - get to the College of Winterhold (I usually use the carriage since a Mage is to squishy for the journey without a great follower or major flesh spells - plus, it's just easier). If you start below level 20, Saarthal is a breeze and the rest don't get too bad until Mzulft.
Just my methodology and it seems to work very well and fit with my RP style. I realize there may be some minor spoilers here, but, I'm can see you're no nooby.
Hope this helps. Sorry for the wordiness.
