I discovered many truths on that run. The first one was that I didn't like playing a healer. (Turns out I just needed to roll a Shaman, but I digress.) The second was that people expected me to heal. The third was that if people expected you to do something, brought you to the instance to do it, and you didn't even know that's why you were there, there would be a lot of upset people, yourself included. So disheartened, I told my then-girlfriend that I was thinking of quitting the game, and she looked at me with that special look between us that means I'm being incredibly dumb and said "Well, you could reroll." This was also news to me.
And so I discovered that you could have more than one character in World of Warcraft. After going over my options, with more of an understanding of what each class was supposed to do, I chose a warrior since he had absolutely no healing abilities at all. Since I wasn't interested in healing (and at the time Alliance didn't have Shamans, so I didn't know how awesome Chain Heal was) and I didn't particularly want to be sneaky or set things on fire, my options were basically warrior or hunter, and being how my girlfriend was playing a hunter, I figured I'd go with the other class and see if I liked it.
I now have five warriors above level 60, three at 70, one at 66 and one at 60, plus two more in their high 40's. So yeah, I liked it. I also ended up proposing while we were running Molten Core, which led to a lot of happy shrieking that confused everyone on Teamspeak, so I'd say that the class selection worked out in the end.
If you find yourself playing a class that you just don't seem to get like I did, you should give another class a try. You may find that you enjoy a paladin's seal and judge mechanic more than a warrior's constant dependency on bring hit, you may find that the stealth aspect of a rogue suits your play more than that mage you first rolled when you started the game. One of the best hunters I know started off as a warlock before discovering that he was much more in tune with his hunter, starting over after leveling the warlock to 60. There's no reason to play a class you find yourself not enjoying, especially when there may well be one out there that you'll find perfectly meshes with your personality and in-game goals. I know rogues who first leveled as warriors, mages who used to be druids, priests who have well geared hunters sitting on the shelves.
If you find that you're not enjoying the character you're playing now, ask yourself what it is you don't like about it. What can't you do that you want to do? Is it just that other people expect you to do something you don't want to? Sometimes that's just their problem. If I'd have stuck it out with my paladin, I'd have eventually discovered the world of protection and retribution spec, so don't be too quick to give up just based on what other people expect.
But if you find a fundamental aspect of the class dissatisfying... if you hate working up combo points, for instance, and would rather not try and arrange to attack an opponent from stealth... then you should move on to considering what you would enjoy. Do you want to deal death at a distance? Okay, we now know you should be looking at mages, warlocks and hunters for your first choices, with druids and shamans also offering you a ranged option. If you don't like the idea of enemies coming over when you attack them and don't want to wait for higher levels to give you ways to prevent that, then warlocks and hunters have pets that can help tank to prevent in your face attacks. Hunters will have higher armor up front and their attacks are physical damage, while warlocks will grow a hardy assortment of fear/horror attacks and damage over time spells to help make up for their lower armor, and of the casters have the most base health since they can gain mana by damaging themselves via Life Tap. If you like the idea of wandering the woods with a boon animal companion, then hunter is probably your option, while if you like the idea of making pacts with demons, you'll go with a lock.
This is just an example, of course... you might well choose to look at other aspects to make your decision. The real point is just that there's no reason to shackle yourself to your first choice if it's not working for you. Keep in mind that the game changes via every patch and hotfix: if I had been more prescient I'd have kept my paladin and used him as an AoE tank once the changes were made to their tanking abilities. Changing mains doesn't have to be a scorched earth choice.
It's a game, after all. You shouldn't be logging on dreading playing it. Take a look at Elizabeth's guide to starting a new class, consider if any of those class' abilities sound like what you wish you were doing, and give it a shot. Better to try something new than to curse what isn't working now.