There is a BSAC dive club that go out regularly. I belive most are off to the Red Sea for a live aboard this weekend.
You can also go to Pearl Divers, it's a shop on Al Nasser St (where Doha Clinic is) and they may be able to give you inof about more PADI stuff.
As for the dive sites..........it depends what your used to and how avide a diver you are. I'll let you find out for yourself ;) Good Luck