The fastest way to get to 80 is, as everyone should know, by doing quests. And quests are best done as a damagedealer. (On a sidenote, you should get two gathering professions as well and keep them up to date. It will help immensely once you reach 80 in term of gold to spend on gear.)
From level 1 to 60, you get double the xp from quests and if you have heirloom items, you get 20% extra experience points from killing mobs as well.
The levels 60 to 70 is also best done by questing, but at 68, you can start queuing for instances as well, since you start getting emblems of triumph from your first random regular Lich King instance of the day. You should also pick up any quests you can get for the instances, as that will be quite a boost in a short amount of time.
Once you reach 80, you should have a decent damagedealing set from both questrewards and by what a crafter can make for you.
You can now start doing heroics. Each heroic will reward you with 2 to 6 emblems + 2 from random heroic (up to a max of 30 emblems a day.)
Now, here's the trick: Hire a tank and/or a healer to boost you for, let's say, 70 gold each instance.
A heroic can be done in 15 minutes, even less, so if you pay a tank 70g for each instance, the tank gets 280 gold an hour, as well as emblems. You get 4-8 emblems for each run, as well as gear. When it comes to tanking or healing gear, you should roll need on whatever is an upgrade to you. The less time you have to spend on getting the gear, the less gold you have to pay the booster and the less time you have to spend to get ready to raid with your character.
So spend 700 gold for 10 runs and you should be somewhat decently geared as a tank or healer, as well as having a tier 9 piece.
For tanking, once you reach 535 defense, you're ready to tank heroics on your own, so you don't have to pay a tank to boost you.
Or you could simply just say "Screw tanking/healing, I'll just hire a tank and get instant queues and upgrade my dps gear instead."