Lots of cool info here!
Starting out, I would suggest try a few, see what fits. I agree with a hunter being a good start, and perhaps a paladin, since they can wear mail and heal. Druids, also, as they are versatile. Cloth wearers are not for the faint hearted, but the strategy involved in working with the challenges are so much fun and pay off later!! End game, there is nothing in more demand for raids and instances than a good holy priest, or a mage with crowd control or DPS on numbers!
However you balance an end game instance team, whether it be for crowd control (priests shackling undead, mages "sheeping" numbers) there are a few classes that are practically indispensable. If you are in a crowd control situation, you have to forego any AoE spells and tame the impulsiveness of charging warriors (are all warriors so suicidal??) and practice tactical play. This situation is also where a badly disciplined hunter becomes one of the most likely to inadvertantly cause a wipeout. So though it may be an easy class to start with, end game they can be one of a teams greatest assets, or greatest liabilities. "I didn't realise my pet was on aggro!" or "I didn't realise I just put aspect on" when your party members are suddenly stunned.
Rogues are extremely useful for DPS, but I question them tanking, being leather wearers! A rogue can stealth and sap, very useful for disabling an opponent. A warrior can stun, but quite often a charging warrior aggros more than he stops. A hunter can use concussive shot and well placed traps, but needs to train his pet well.
There are only two classes, IMO, that are capable of truly soloing certain areas, and they are the rogue and the feral specced druid. I have a level 66 feral druid, she has 13000 armour and 8000 health in dire bear form, more if buffed. At level 58, with two other druid friends, we stealthed Black Rock Spire in cat form just to get the Wildheart boots from Mother Smolderweb
It took a few of runs, because of the drop rate, but each run was only 20 minutes. I have read of this being soloed by a druid.
I have healing gear I can wear, but at this level am not a main healer since I am a feral specialist. Cat form for stealth and DPS (176 DPS, able to crit 2500), bear form for tanking and holding aggro, (can take on 3 same level with ease, or more if a priest is backing up health). Improved Leader of the Pack, a feral ability, will heal party members for a percentage of their life each time they crit. So feral druids play tank, off tank, DPS, or back up healer. I always have to be sure of my expected role when invited to party an instance. Oh! and warriors have a beef about druid tank abilities, they got us nerfed in the last patch, but not a lot!
Another note in favour of druids...at level 68 they get flight form. No druid should spend the 1000g on a standard flying mount, as we have proved it is no faster (unlike the level 30 travel form being slower than the level 40 mount) Also, because of their flight abilities, they get automatic training, which saves a 1000 gold off the cost of the epic flying mount! But level 70 is a long way of for you yet, don't rush to get there, there is much to see on the way!
Like I said, play a few, see what fits you!