Ambergris Caye, Caye Caulker, Placencia are all good choices. The two indicated airlines (Maya Island and Tropic) have flights to all 3 from the international airport on little 15(?) seaters; if you're into it, you can ask for the co-pilot's seat which I've gotten a few times.
Note that if you're looking for peace and quiet, in and near San Pedro is not the best choice, but far enough out of town would be fine; lately I'm finding Caye Caulker preferable, and will probably give Placencia a try next trip.
You can book fishing trips out of San Pedro. Not sure about Caye Caulker, but that's only 20 min by boat from San Pedro.