One of our main roles as a protection paladin is carrying the flag. We're pretty hard to kill, have a range of defensive cooldowns and can even help the healers out with self heals. I'm going to be talking about this from a random BG perspective so it might not all be applicable to rated BGs. This is also all from am Alliance perspective as I haven't played much horde side.
1) getting the flag. If you're in a random group, you're going to want to get across the map and nab that flag ASAP. I find the quickest way for both flag carrying maps is on the right hand side as you come out of your base. For Warsong head straight across the map and up the outdoor ramp, then through the first door you come to and drop down on the flag. This has the advantage of sometimes not dismounting you until you are right at the flag, and even if you are dismounted as you go inside, you should still easily beat anyone coming up the tunnel as you are mounted for much longer. On twin peaks I like to stick to the right and then cut across the rocks to head for the front door. While this route is longer than the direct route I find I very rarely encounter any resistance. If you have a DK with you, you can stay right and go straight through the water.
2) Flag picked up. The most important thing from now until you cap is keep your healers in LoS! Some healers are good at sticking with you and anticipating, but don't expect them to off the bat. It is better to take combat with your healers than get completely isolated and then get hijacked by a couple rogues and slaughtered.
3) Abilities. Hand of Freedom is brilliant defense against druids and mages snaring you the whole time, make sure its key is near to hand. Guardian of Ancient Kings, Divine Protection and Ardent Defender are all good for reducing incoming damage, and AD even base a free life attached, but if it's triggered you're probably in trouble anyway unless you get a lucky Lay on Hands. Avenger's shield and Hammer of Justice are great for slowing down attackers (I assuming you're glyphing dazing shield). And try get your CS / HotR hits in on anything, even pets. You want you HoPo for Words of Glory.
4) Healing. Word of Glory is a great asset. Pop wings if you can to increase the heal. Lay on Hands is the greatest free reset spell you could possibly hope for. The only thing that would make it better would be I you could cast it while incapacitated. Get the glyph and pop it if you're in trouble and isolated.
5) Beware. Don't get panicked and bubble. This will drop the flag. LoSing enemy casters is a good thing, but make sure you don't leave your healers out of sight as well.
6) Keep an eye on the enemy flag carrier. The default UI will tell you who is carrying your flag. Try target or focus them or watch your raid frames (you do have raid frames that show targets right) and when you see the EFC going down start heading for the flag spawn spot. There is nothing worse than going to all that trouble of returning a flag, only to have it recaptured because your own FC wasn't paying attention and didn't cap.
Finally when you can cap, CAP. Don't be that Guy that stands in the flag room prolonging the battle unnecessarily so that you can "farm kills". Win and requeue, the queues don't take that long, and also it's just rude.