Tell your team you are a healer
Before the gates even open, tell your team that you're a healer. Announce yourself and your general style. Tell them things like:
"I'll stick with the flag carrier to keep them well-healed."
"I will go with a small strike group to overwhelm enemy defenses."
"I will stay on defense."
This sets some expectations with your group. Since you'll want your team's protection during the match, it's fair to tell them where you'll be and what you'll be doing. This is one of the most important things you can do to encourage cooperation throughout the rest of the game.
Pick a partner
You have the option to join up with a pocket DPS character. Work out with your DPS partner whether you're taking the lead or he is. If he's going to charge around and pick out targets, then you simply need to follow them. Conversely, if you're taking the lead, you need to trust that your DPS partner will stick with you and protect you.
Trust me when I say that the DPS character will be busy. It's tempting to believe that running healer guard duty isn't a full-time occupation. What you'll quickly discover is that every rogue, druid, and other character will take pot shots at you and make assassination attempts on you. Your guard will need to deal with these attacks quickly, keeping you free to throw heals out to the rest of the team.
Your friend, the raid frames
The raid frames are your friend. While plenty of user addons can modify the game for healers, I consider the current in-game raid UI to be sufficient to the task. Nearby players are highlighted, while out of range players are dimmed.
Clearly, your first focus will be nearby healers. At a basic level, getting heals out to your team through the basic raid frame works much like PvE. If you see someone with low health, throw them a heal. If you can cleanse, go for it. Just playing this kind of whack-a-mole is more than enough to keep you occupied through most of the match.
Where expert skills come into play is both watching the raid frame and the action happening around you. Picking out which team members have become the target of multiple opponents and therefore in the most need of your services is the real art of healing in a Battleground.
It can be tempting to keep your eyes fixed on the raid frames at all time, so do your best to fight that urge. Look at the immediate area around you and pick out where you can see a group of opponents. Click one of your enemy, then hit F to find out who they're targeting. That person is probably the most in need of your heals.
You should keep yourself among your highest healing priority. Sure, extenuating circumstances will sometimes demand that you heal the FC first or something like that. But as a general rule, a dead healer throws no heals.
Once you've got that priority out of the way, start getting tactical about your healing. Your decision to heal your team members make your heals a weapon. If a warrior is doing an amazing job killing other members, your heals become an extension of that warrior's DPS. If you want to make sure that warrior's target dies, provide him extra heals until the target is toast. In this way, your heals come a sort of DPS in their own right, because you select which flows of DPS (characters) continue.
Get all judgemental
This is the toughest part of being a healer; you can't be everywhere at once. You can't be next to every player. You have to pick and choose. Which team members are worth healing and which are not?
It's hard to give good, universal advice about who deserves heals and who doesn't. That's got to be up to you. Figure out who is putting in effort for your team by doing things like defending flags. Ignore the people who are fighting in roads. Outside of those few basic tropes, though, everything is one big gray area.
But if you try to heal every member of your team, you will lose and you will not heal effectively. Figure out your criteria and stick to them.
Visit the WoW Rookie Guide for links to everything you need to get started as a new player, from how to control your character and camera angles when you're just starting out, to learning how to tank, getting up to speed for heroics and even how to win Tol Barad.