Well, obviously this is why trips aren't as strong as a set. Everyone sees those two 6's out there, so if you get significant action, you're likely in trouble to either a boat, or a better 6 (Did they really limp with 62 or 63?).
I would classify this situation as a way ahead/way behind scenario. If no one has a 6 with you, then you're in great shape, and they're drawing to 2 outs with an overpair. If someone does have a 6, then you're the one looking for a three outer.
You're not likely to get a good outcome from a big pot, so you're going to want to check either the flop or turn. I like checking the flop here, trying to let someone catch a piece on the turn enough to pay you a small bet or two on the turn and river. Don't try to find out where you're at through bets, you can already do so through analysis like I have just conducted.
Check either the flop or turn, and bet the other two streets modestly, whatever you deem you can sell it for, but don't be looking to play a monster pot with this hand.