After finishing up the Arena Tournament, I figured I'd share some of my hard won knowledge. My team did pretty good, 20/30, and I found it to be a wonderful learning experience.
I played a disc priest. It's the class I'm most familiar with, and is generally considered to be pretty decent in pvp. Of course, my personal survivability in BGs has always been rather stinky, because I tend to approach BGs like I do raiding. I heal everyone and hope the big bad/many small bads don't notice me. This is NOT an effective approach to arena. It's not really an effective approach to PVP in general.
So, my first lesson? Your survivability is your own responsibility. A dead healer heals no one. Disc priests have a lovely amount of tools to ensure we survive. We have our overpowered shields, several different heals, Pain Suppression, and a fear. You need to look at what your class has to help you survive and learn how to use those abilities.
Second lesson, shutting down your opponents. Once you get the hang of survival, you can move on to annoying others. There's some overlap here, as many of the abilities that help you survive do so by shutting down your opponent. A fear's a good example of this.
Third lesson, killing your opponent. For a healer, this is always a bit of a challenge. Let's face it, healers are not big in the damage department. But every class has some damage-dealing abilities we have access to, even if they're not very powerful. For a disc priest, you can smite, shadow word:pain or death, use devouring plague, holy fire, penance, and even that aoe heal/damage spell that I can't remember the name of. I'd suggest running around the world for a bit and killing mobs with various spells, to try to get a sense of what's going to be the most useful/powerful for killing. You can also try practicing on a target dummy.
Fourth, know your abilities. This is the biggest lesson of all. Stuff that you never use in pve, stuff you thought was useless, well, that stuff can be highly useful in pvp. Take Mana Burn, for example. It's not a spell I've ever used in pve, and in fact I didn't even realize I had it till someone pointed it out. But in pvp, denying your opponent mana can win you the match. Drain a warlock's mana pool and either they have to try wanding you to death or they life tap and make themselves that much easier to kill.
The nice thing about the Arena Tournament is that the gear is equalized. Everyone gets fully kited out in pvp gear, with access to all glyphs, trinkets, gems, and enchants. In regular pvp, it's gear+skill that determines the outcome. If you don't have enough resilience, it doesn't really matter how good you are, you're going to be one or two shot. So, if you want to get into arena pvp now, I'd say run BGs, gear up a bit, and then find an Arena team. And even if you're not interested in competitive pvp, pvping is a great way to learn more about your class. Heck, it might even make you a better raider, since you get a much better sense of your class's oh shit buttons and it teaches you to think on your feet.