I certainly wouldn't call it 'great' diving, but when I lived there serveral years ago friends and I would hop into Offats Bayou on 61st street. Poor visibility and shallow depths - around 25 feet max. We would use BBQ thongs to catch crabs (mostly blue, some stone crabs).
Other than that, I think you'll actually need to hop on a boat to get offshore a few miles to do anything 'great'. I've been on rig dives out in the Gulf which are fun.
It's been several years since I've lived there so I'm not sure I can provide good info on shops. The only one I knew of was The D.E.E.P. on Seawall blvd, but I think it shut down a few years ago. Others on the board may have more info on dive shops / boat charters.
-Jeff