First of all, welcome to WoW, we will enjoy you crying over the addiction playing with us and everyone else, and we hope you love the game just like we do.
Just a heads-up, you meant "What's the best race for a mage", not class, because mage IS the class, whereas gnome-human-troll-etc are the races.
1)There is no "Best race" for ANY class, there is the argument of some being better than others because of the racials, but they are really the minimal part when making the decition. Sure, if you dont give a rat's a$$ what race you are, then yeah go with the better racial's one.
If you are one of those people, go with UD mages, for Will of the Forsaken, BE mages are also nice for the enchanting bonus, and the bonus silence racial.
As far as alliance go I would say go gnome... they are just amazing, and their escape artist racial is just so good now its a great bonus to the already cool race.
2) You learn your spells just like every other class does, at your class trainer, found in most major cities, excepting shattrath, and in most( If not all) of the... lets say... "secondary newbie starting zones' towns" found after the starter area (These are Goldshire,Kharanos, that wierd D-starting named town for elves, and the "Idonknowwatzitcalled" place for draenei for alliance; Raptor Hill, Tranquillien(sp?), Brill, Bloodhoof village for horde).
These spells cost copper,silver, and gold, depending on what level is required to cast it, for example you will be able to learn frostbolt rank 1, once you get level 4 and pay one silver for the spell. Then once you are level 60, you will be able to learn frostbolt rank 11 for 4 gold and 20 silver, it will be the same spell, but more powerful, and will cost more mana.
3) Definitely the best way to make money is the always mentioned "grab 2 gathering proffessions" wich means grab proffessions such as (dis)enchanting/herbalism, herbalism/skinning, mining/(dis)enchanting, mining/skinning. SELL all the stuff you get, and you will be rich in no time.
Also, dont buy new gear at the Auction house every 2 levels, just upgrade your gear if it is at least 12 levels below you, and even then its not necessary sometimes.
(Besides gear) avoid buying stuff at the auction house unless it is something you know you will have for a long long time.
As far as getting armor, quests will give you very good armor (and weapons too), and as you progress in the game instances will give you the best (or almost the best) gear available for your level.
Rarely you will find better gear at the AH, but dont buy any unless its really cheap and you know you wont upgrade it in a decent amount of time (10 levels or more)