Obviously, it's important to play well, but a healer's ability to communicate well is equally important to the ability to push buttons at the right time. Raiding is all about teamwork, so if you can't communicate effectively, you'll have a much harder time working with others than if you did.
Priests in particular have many things we need to talk about in combat. Pain Suppression, Guardian Spirit, Power Word: Barrier, Hymn of Hope, Divine Hymn, and Leap of Faith are all abilities other players need to know are being used, so let everyone know when you're using them. You especially want to communicate their use if you have to (for whatever reason) break the original order you were assigned to use those abilities. You can use /say and /raid macros that tell players what you're doing, but it's always best to back up those macros with a comment in voice chat.
There are several other general things to communicate in a new guild as well. First, ask questions on fights you don't have experience with or don't know the strategy for perfectly. ("How do you guys spread out for this air phase?") Each guild does things differently, so knowing exactly how something is done will save you embarrassment. If you do make a mistake ("Oh my god, I just lifegripped the tank! I'm sorry!"), you should always admit to it and apologize, especially if you've caused a wipe. Doing so will make you seem accountable (which most people like in a raider), and if there is any way to salvage a mistake, sometimes you can do so if you're up front about it.
Finally, ask what your healing assignments are, so you know what the guild wants you to be doing as a healer on each fight.
Don't let your target die
Once you know who you're healing, it's paramount that you keep that target alive, no matter what. Prioritize that assignment over all things (except your own health -- you need to be alive to heal). You can heal other players outside your assignment, too, but if you can't comfortably switch off the tank, off tank, or whoever you've been assigned to heal, then don't. There might be a juicy health deficit in group 3 that hasn't been touched because the player was temporarily out of range, but you need to always prioritize your target. If your target dies, your guild will think that you're either incompetent or you care more about healing random rogues in group 3 than doing your job. Neither are desirable.
Should you fall behind on your target for whatever reason (healing that rogue in group 3) and you suddenly find yourself unable to keep your target alive, remember that you are a priest and have abilities like Pain Suppression and Guardian Spirit. Use them liberally if you have no other way to keep your target from dying. After you've done so, remember to communicate. Tell your raid, "Hey guys, I had to use Guardian Spirit just now, so mine is on cooldown now!" This can create problems in some fights, but for most fights, your group can usually juggle around cooldown abilities and work through it -- but they'll only know to do this if you tell them. The end result of this is that the guild will know you know when to think for yourself to keep your target alive (even if it's your mistake that he needed the cooldown in the first place) and are able to resolve any problems that come from changing the plan mid-fight.
Numbers do matter
Everyone knows that meters don't matter so much for healers. Logically, we know that a player who lands a Flash Heal on a tank when his health drops to 1% is more beneficial to the raid than the player who ignored the tank to cast Renew on five players who had a 8% health deficit. (Sure, those people have to get healed eventually, but the player at 1% should take priority.) We also know that the player who consistently chooses what's best for the raid is better than a healer that obsesses over meter performance and is always at the top.
But even though we know that, we ignore it a lot of the time. We ignore it because it's fast and easy to use a meter to judge someone's performance. If someone's healing isn't high enough, they must not be skilled enough or not trying hard enough, right? Not to mention, trying to assess someone on actual performance takes thought and effort -- and you have to pay attention! Pft! Forget that!
That's why meters matter when you're doing a trial. There will be tons of players, both members and officers, who will assess you with a quick glance at a healing meter. If you're sitting at the bottom, they will think less of you. So try your best to find a medium between greedy healing and helpful healing. Always keep your target assignment alive, then milk effective healing numbers from any target on which you can get a heal off. Frostheim would remind you that guild trials are a great time to heal hunter pets! (He'd also remind you that every time is a good time to heal a hunter pet, but we won't go into that.)
As a priest, you can of course find many ways to make sure you're getting lots of additional throughput. Use Circle of Healing on cooldown, and keep Renew up on anyone taking damage (as long as you can spare the mana). Use Greater Heal instead of Heal, just in case your target loses a tiny bit more health before you get your cast off. Always make sure Prayer of Mending is active and on cooldown. Set your boss timers to let you know whenever big raid damage spells are about to be cast, and time a Prayer of Healing just after or a few Power Word: Shields before so you can get full credit for the healing done by them.
Find your healing niche
Every raid group that is looking for a new healer has a hole to fill that is more than just a player vacancy. You might be getting picked up because you have Power Word: Barrier, but you aren't just there to cast that spell every 3 minutes (particularly if you happen to be holy.) Other healers will have things they naturally gravitate to doing when they aren't healing their assignment. Druids will often stack HOTs on players they think will take damage. Shaman might use Riptide on any random player who needs a heal. Paladins will be checking their hair. (For some reason, the expression "holy check" just fell apart in my brain.)
Your job as a new recruit is to find out what job isn't being taken. Is there a player in group 2 who never gets any heals cause everyone hates him? Make him your new best friend. Do the healers get too busy to heal themselves and die during phases with strenuous raid damage? Heal them!
Watch your raid frames carefully, and look for trends. Even if you don't necessarily know who is healing what, you should be able to see a pattern of what people tend to heal and not heal. Your job in finding your niche is to heal what isn't being healed. And if that's not a juicy enough assignment for you, then you can say "Screw finding a niche!" and try to do someone else's job better than they can (you smarmy bastard). If you do that, you'll need to be fast in switching between your assignment and everything else. Don't screw it up!
As a priest, play to your strengths when looking for your niche. If no one likes healing the tank, then arrange your gear, talents, and glyphs around tank healing (pick up Surge of Light, for example). If no one heals the raid, then make sure you've got all the spirit you can possibly get and go heavy with AOE healing and Renew (or shields, if you're disc). If you're one of two holy priests, try using the Chakra stance the other guy isn't and get what he misses as a result. If you're one of two disc priests and he's a shield spammer, stack up some haste and use Prayer of Healing more.
Ask what you can do to improve
My last bit of advice isn't so much about playing your class as it is political advice. It also happens to be more healer-specific than priest-specific, but don't mind that.
Because raid teams are delicate little ecosystems with relationships and structures that have existed long before your arrival, it's important to respect the structure of a raid team if you're to become a part of it. The best way I've found to do that is to ask your peers what you can do to improve. This will achieve a couple of things. First, it will help send the message that you care about fitting into your environment and growing as a player in general. Second, if you do have any problems, it will allow you to explain yourself (if you know what you did wrong) and identify any issues you didn't know were there. Asking in general will also help you figure out what the guild is looking for from a healer and help focus you on improving or emphasizing those traits. It also has the bonus of making you seem more amicable and thus make it easier to gain friends.
If you're not sure who or when to ask, go with your gut. I personally like to ask a player I respect, be it a healing lead or a good healer in the raid. I'll ask once a week, before or after the last raid of the week, depending on schedules. (Remember to be understanding of the person's time!) If you ask an officer, you'll have the added bonus of getting a person who might be voting on you later to think about how you've played. If you played well, they'll remember it better later when you lifegrip the tank. (What, you never lifegripped a tank?)
And with that, I will leave you again this week. I hope this was helpful to any of you starry-eyed raiders looking for a new home. Next week, we'll be moving onto guides for patch 4.2 and the Firelands. Let me know in the comments if you have any questions in particular you want covered, and I'll see where I can tie them in.
Spiritual Guidance has the inside line on pre-raid gear, valor point and raid gear, and healing strategies for bosses such as Atramedes and Chimaeron. Newcomer to the priest class? Look into Discipline Priest 101 and Holy Priest 101.