It is said the priest is weak at the beginning since they cannot take or deal lots of pure physical damage. However, many people favor the profession a lot. We have ways of dealing with or avoiding the damage, and if properly, we can deal lots of non-physical damage. You don't wear bulky armor or carry oversized, cartoon-like weapons. Priest damage and survival is more elegant than that. As for being weak, once you reach level 70, a properly played shadow priest will be at the top on damage charts for a long time.
For most of us, being priest, we are smitten with their task—healing. It is said unless you have a regular group with which you are going to party, you will probably embark on healing on your way to level, regardless of whether you choose to be shadow healing, holy healing, or discipline healing. Get a feel for healing, and see if you like it. There are 3 healing classes in game if a healer you wish to be. Firstly the Druid, who can heal and also transform into a bear or cat form, which allows some impressive damage alongside some quality healing abilities. The druid can later re-specialize for a restoration build which is very efficient at healing a group. You can easily solo Druid character and still make wow power leveling easily.
Shamans are races that will do massive damage per second as a melee or spell cater specialization yet have fantastic healing ability? He/she can also re-specialize later to become a very efficient healer for a group. You can also easily level a Shaman.
As for Priest, they have some great attacks as a Shadow specialization and are feared by all, as a spell caster. But if the priest is specializing as a holy healing, there is no better support class in the game. I mean it is cool when you have to keep track of the health of your party members, and constantly make judgments on the various heal needed at specific times. It's probably one of the cooler classes to play in that regard, as instead of button smashing, you actually have to make intelligence on the spot decisions to keep the party alive and functioning.
Also, as a priest, if you're with random groups which you will often combined with in the game to fulfill objectives, you have a good chance of taking a lot of craps from some arrogant jerks who can't play properly and are looking for someone to blame for their own shortcomings. If the group completes a major quest, they'll forget about how you brought them all back from the brink of death and give glory to the tanks.
So if you love to be a priest, you can go on with it, which will surly satisfy you and make your WOW tour more wonderful.