What they said with a little fine tuning. Some of the shops with boats in the North Sound also might go east at times, it depends on demand and schedule. Many need the boat back for Stingray City or the Kittiwake also so that limits how far they'll travel on a 2-tank. In addition a few offer 2-3 tank extended trips that often go farther east - and include lunch.
Additionally some shops keep a boat near Georgetown - that may go south or as far north as West Bay
The best thing to do is probably call a few shops and find out their site plans and see if those are dives you want to do. Tortuga or Ocean Frontiers regularly dive east side sites like Babylon but the gas/time make that prohibitive for the west side shops unless they do a 3-tank extended trip - often with lunch included.
www.idivecayman.com has videos of most of the popular sites.
Don't forget to shore dive if you can work out a buddy - CITA likes you to have one. A good day of diving is start at Cobalt Coast, go out to the North Wall (Reef Divers might be able to drop you off) then move west and dive Turtle Reef followed by Lighthouse Point. The Kittiwake is also close by and most shops have a fixed appt.. so find who does when you want to dive it - if you do.
Reef Divers is on-site at Cobalt, SunDivers at Turtle Reef. You can reach Ollin there thru Macabuca's website - there' a page on it. Divetech is at Cobalt Coast - conditions permitting they'll dive Ghost Mountain about 10misn by boat from their dock - worth the drive up.
I'd personally avoid downtown, the dives there are tired due to the cruise port and last year a container ship collapsed one of the Eden Rock caverns. It's really only good when the Silversides are in and you just missed it. Some days up to 4-5 cruise ships are in town so if you're driving thru, they're looking the other way b4 stepping off the curb in front of you...
Suset House is worth a look one afternoon also, easy entry shore dive, the Mermaid stature, the Church photo school/gallery/store and My Bar has a view and good food. It's one of few places where finding a buddy is easy since they don't afternoon boat dive from the resort.