Ok, well, first off, if you're starting PvP, you should do random arenas. I know it's better if you can play with a couple of friends in the team arenas, but frankly, many of the teams there are very experienced, and will probably cream you before you have the chance to learn anything. So, head over to Lion's Arch (or even earlier, if you want, although some of those arenas have twink problems that are more frustrating), and start up.
Now, here's a brief crash course in arena PvP. It's not meant to be that comprehensive, and while some of the general concepts do apply to team arena or GvG type, many do not particularly, so you will have to adapt as you see fit with the new types.
First off, preparation. I'm going to assume you're at Lion's arch, at your present level of 18-19. Since you have access to the ascension missions, it is probably worth your while to try them for some elite skills. For instance your E/Mo will greatly benefit if you can pick up ether renewal skill that I believe you can capture at Thirsty river. For arena fighting, if you don't have Draknor's armor, you're at a disadvantage. A fairly significant one. It is disadvantageous to be under level 20 in this arena, so you might want to see if there are a few quests you could complete to try to level up, but even at level 19, or 18 if you're careful, you're still pretty competitive. Just remember, you take more damage and deal out less to a higher level player.
Next, skills. Because you're in a random arena, you may find yourself in a party that is in some way deficient in certain areas. Most significantly, you might not have a healer. Since you can't really assume you will, you should grab some form of self healing. You should also grab a resurrect of some kind. Be mindful of the casting time though: while restore life may look really tempting, the fact that is a touch spell AND has an 8s casting time, makes it pretty hard to use effectively. If you can, particularly if you're a caster, you should try to grab one skill that does energy management (ie, some way to get back lost energy). There's nothing worse than having your monk run out of heals and have to wait for his 3 pips of energy regen to fill up so he can get a single cast of orison of healing.
Next, look for a niche. Try to figure out what it is that you will be contributing to your team. These fall into a few different categories: spike, disruption, healing, protection, damage over time (DoT), area of effect (AoE), etc. What do these terms mean? Well, spike damage is like getting hit by a falling piano. You get hurt really bad really fast. Disruption means using hexes, certain types of attacks, etc, to try to keep your opponents from attacking properly or using skills. Healing and protection are exactly what you would expect them to do. DoT refers to either healer drainer type characters, or your basic tank/ranger type who just keeps dishing it out. AoE uses skills that try to damage multiple targets at once. But what should your specific class do? Well...
Warrior: You have two basic jobs. One is to absorb lots of damage from the enemy. The other is to kill enemy monks (and mesmers, to a less extent). Since you do a lot of damage, the monks will probably try to run away from you, so you want to bring skills that improve speed, or better yet, slow the enemy down. If you use a hammer, take knockdown skills. For sword or axe, take high damage skills, and don't forget things like cripple or disrupting chop. Any status inflicting skills are always a bonus. I wouldn't take too much healing, even as a W/Mo. A healing signet or something similar is probably good enough. If you are a W/Mo and bring healing spells or especially enchantments like mending, you should put them on your monk.
Ranger: Rangers are either disrupters or spikers. They do very well at both. If you're a disrupter, pump up your expertise and marksmanship, and take skills that cause interruption. Distracting shot, punishing shot, etc. are good examples. If you have something like apply poison, crippling shot, etc. these are also worth having for extra damage. Spike rangers will need to probably look to their secondary for help to deal extra damage. In particular, if you're a R/Me or R/E, there are a lot of good options for dealing tons of damage. Check around these forums for various spike builds. As a ranger, you should *always* be targetting an enemy monk or mesmer.
Monk: If you're a monk in a random arena, I recommend you be a healer. Healing monks are everyone's best friend. Your job is to keep your team alive, and, more often than not that means keeping yourself alive. Any decent team will try to kill you first. Be prepared for it. Bring lots of cheap heals with low casting times (orison, dwayna's kiss, mend ailment is good for conditions even if you have only 1 or 2 points in protection), something like healing touch for yourself, and if you can find the space, grab something to try to keep a warrior off your back. Ideally, you should be able to tank the average warrior without having to run or waste too much energy on yourself. That doesn't mean you *should* tank the warrior, but having that option is a bonus. Try looking into your secondary. In particular, anything that does slow, blind, or weakness are very useful. Don't worry about damage, avoid things with high energy costs or long casting times. If you have a good energy management system (I use offering of blood with 10 pts), then you can actually tank the warrior and heal your party without needing one of these spells, as long as he isn't using a hammer or certain other builds. Once you get more experience, you might want to look at some protection/smiting builds as well.
Elementalist: Eles can be tricky class to play in PvP. A lot of the spells that are the elementalists bread and butter in PvE aren't very useful in PvP. In particular, spells like Meteor Shower, Firestorm, Earthquake, etc. are out. Ideally, you want to avoid anything that causes exhaustion, because it can really cripple your energy management. That said, you still have a couple of options. If you focus heavily on earth, you can act as a protector, using wards, slowing hexes, etc, to gain the upper hand. If you take water, you'll be playing as a hexer, trying to use your debilitating water hexes to confound your enemies. Will, maybe annoy, at best. If you play fire, you'll be looking more toward AoE. Sometimes these spells are useful defensively, like if you drop a searing heat or firestorm on top of the guy attacking your monk. They can also be used offensively, as either AoE or spike damage. Try to take ranged spells with medium to low cost and short casting times. As usual, your main job is to try to take out enemy monks, or use AoE as a disruption to their healing. But, in my opinion, if you really want to be an elementalist (and actually use elementalist skills--smite monks are often E/Mo) in PvP, air is the only way to go. These spells are fairly low cost, fast cast, many can strike without a projectile (good for killing runners), have armor penetration, goodd damage, and have some nasty disruptions or conditions associated with them. Take things like enervating charge (best air spell in the game IMHO), whirlwind, lightning strike. Attack enemy monks/mesmers from a distance. If you feel like it, you might want to try combining with earth for some interesting combos. Whirlwind/aftershock and enervating charge/stoning come to mind as good options. If you have ether renewal, you can also use stone daggers as your spam spell. Finally, elementalists can make decent healers (E/Mo at least). Max out your energy storage and healing prayers, grab ether renewal and a bunch of healing spells. You don't have to worry so much about higher energy ones like heal other or heal party, since your energy management far outstrips the monks you'll be competing with.
Mesmer: You have basically one job. Disrupt, hex, and otherwise piss off the enemy. Grab energy steals, backfires, etc. for enemy monks. Bring something along for warriors too, since you will be a major target. Don't bother with empathy for warriors, get something that does blind or reduces hit % or even slow. Try to stay on the sidelines and make life miserable for the other team.
Necro: Necros are tough in PvP. Particularly in arenas, there aren't enough corpses for you to really benefit from those types of spells. Focus on curses or blood magic probably, particularly either heal drains or things like mark of subversion. As with the mesmer, your target is mainly to wreak havoc on enemy monks. Bring something like chillibans or rend enchantments, too (careful using this last one!) Alternately, fill your skill bar up with condition skills, and try to use these skills to decimate your opponents.
That's all I can think of for the moment. The best way to learn PvP is to play a lot, and have fun doing it, really.
Rico