I must be missing something obvious, but I can't figure out how to tell if I'm actually in a rest area or not. Is there some buff or tooltip I'm missing?
Anywhere in your ship is a rest area. Most parts of cantinas and in the fleet are rest areas.
You can tell when you enter and exit them by a screen message, and an ops message in your log.
But if you missed the enter message, the easiest way to check if you are currently in a reset area is to mouse over the experience bar and leave your cursor there for a bit. If you are in a rest area, it should start ticking up your accumulated rest amount fairly quickly. Earlier levels not as quick, but once you reach level 15 or so the rest should be ticking up by 1 every couple seconds or so. (That's just an estimate; the point being it doesn't take too long for it to increase, if you're in a rest area.)
As far as I know, there aren't any persistant UI indicators for being in a rest area, unlike some other MMORPGs — if that's what you were hoping.
UPDATE: Since free-to-play has been implemented, it has been changed in that free-to-play users do not receive rest XP. Therefore the "rest XP" numbers won't increase for you until you become a paid subscriber. You will see a message to this effect when you mouse over the XP bar in a rest area.