The art of the grab
In the hearty days of yore and lore, a clever FC could sneak into the enemy base, grab the flag by one's lonesome (if there were no defenders), and jet across the field to great success. Moving like a well-oiled lightning bolt, the FC would grab a flag and be in home base before anyone was the wiser.
Those days are over.
Here's why. First, the minimap now shows everyone where you're standing. It takes a few seconds before an enemy FC is lit up, but within a few moments, the entire game has a handy HUD for where to send the pew-pew. You're not as sneaky as you think.
Even with resilience, damage is huge. A lot more crowd control happens tangentially to damage than ever before. (We're looking at you, frost mages.) So if your run isn't crystal-perfect, you're just not going to rocket back to safety before the enemy team tears you open like a child with a Christmas gift.
The modern grab tends to be a small team affair. Two or three folks run inside the flag room, snatch the flag, and then jet off to safety. That safety could be a defensible position, or it could be a spot inside your friendly zerg. But the rule is: Get flag, get to safety.
Get to safety
This is where the trick comes in. You need to discover quickly where the safest spot is. Many FCs advocate just getting up to the friendly base as quickly as possible. That can work, depending on how the match is going, but I generally advocate sticking near some defenders or a defensible position.
By the time you've got the flag in your meaty little hands, you should know who your healer is. Look for them on the minimap. (Setting them as your focus ain't a bad idea.) When you find your healer, check out their status. Are they getting jumped by half the enemy team? That means that's not your safe spot.
You can generally run graveyard, ramp, or tunnel. Tunnel has the benefit of being a direct route, with the possibility of a speed boost along the way. (I always leave that for my healer, to make sure she can keep up with me.) Ramp has the benefit of not being the tunnel, which is where most FCs go by default. Graveyard has the benefit of being a crazy way to go, since that's where enemy players respawn, and nobody expects you to go that way.
In general, people take tunnel. Look at your minimap and be aware of how the game's going. If half your team members are at the mouth of the tunnel, quickly losing health and piling up debuffs ... then that might not be the best place for you. If you're confident your team will wipe out the bad guy, go ahead and join them. But if you're losing that skirmish, get thee to the ramp.
Pay attention to your minimap
You see a pattern in this advice? Look at the minimap. Avoid places where your team is losing fights, and go to places where your team is winning. This is the safest way to cross the field.
When you make it back to your base, you have three basic hiding places: the roof, the balcony, and anywhere else you can break line of sight.
The roof has a few advantages. First, if the enemy chases you up the roof, they are forced to go up that long ramp. That gives your friendly characters approximately three and a half days to blow the bejeezus out of the enemy. Second, if you hang out at the far wall blocking line of sight, you can still tilt your camera to watch the roof entrance for bad guys.
The balcony also allows you to block line of sight, and it shares some advantages with the roof. You can jump down as a way to get out of danger while still allowing you to remain relatively near the flag. The balcony is a relatively restricted area, which means the long distance of ranged characters is somewhat mitigated.
The third option is everywhere else. Everyone seems to have their own favorite hidey-hole. Hopefully, folks will share those in comments. I've seen Horde hide under the saw, Alliance try to find a friendly bush, and any number of other combinations. In general, in a PUG Battleground, I strongly advise against clever hiding locations; if your teammates can't find you easily, they can't help you.
Find your friends
While it might seem like stupid, unnecessary advice, I can't stress how much finding your friends is key. Find friendly players who have the guns to defend you as flag carrier, and get to them. Individual performance makes a big difference in Warsong Gulch, but you're not Superman. Get to friends and stay by their sides.
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.