I felt much the same way starting this game as it was my first massively online game and I didn't know how the social environment would factor in when I wasn't always on as much as others, etc. Anyway, three years later I can say it was a pleasure from the start and continues to be a passion in my life :)
As Defaulty mentioned, WoW is geared toward the new players (as are all of Blizzard's products). It is almost impossible to NOT succeed in leveling up and experiencing all the areas of the WoW world. The quests lead you from one place, trainer, etc. to another when your level needs it. You can play at your pace in a solo mode or you can group up with others and either works to let you progress.
As far as which characters to play, you already mentioned your initial choice is the priest and that will work out fine. It is not the easiest choice for survivability, but all classes work early on. I agree with Defaulty in the "keep it flexible" kind of mindset early on. It really helps to try out many classes early on where leveling is easy and quick. You get a sense for how they all work and that will help you down the road when you are grouped with them. At some point, you will want variety too. This is why I play a priest, warlock, and rogue. I get to heal, range dps, and melee depending on my mood.
Good luck with the priest leveling and don't sweat the small stuff. Apply your same real life social skills to the game and you will find "friends" in WoW soon enough ... and that is where the real draw to this game comes from in end-game play.