It's easy to island hop from Nassau. Not particularly cheap though, I think BahamasAir runs about $160-180 r/t to Freeport.
But there's BahamasAir -
Fly to The Bahamas on the National Flag Carrier Bahamasair. Service to all major islands including Nassau, Grand Bahama, Abaco, Andros, Exuma, Eleuthera, Cat Island and all of the Islands of The Bahamas
Sky -
Welcome to SkyBahamas - Delta subsidiary
Western -
WesternAir
If you do the Nassau thing, Orange Hill or Compass point.
Cove's bundles a package deal with both. I wouldn't call Compass Point particularly reasonable though:
This small upscale resort is located on the ocean and the closest to the main Stuart Cove's Dive Bahamas facility.
Stuart Cove's Dive Bahamas Shark Diving Bahamas Diving in Nassau Bahamas - Dive Packages
Check them carefully too, I once priced a Cove's package separately using discounted rates on Expedia and it was actually cheaper.
Neither are in Nassau though if that's important - Cabs aren't cheap so to get around I'd want a rental car- the Cove's shuttle will pick you up there daily for diving.
An off-beach option is the Comfort Suites on Paradise Island. It's owned by (and view blocked) by Atlantis so you get Atlantis privileges while staying there. There's also a Best Western Suites that might be slightly cheaper.
Atlantis is pretty nice also...lol
Atlantis Paradise Island Resort | Caribbean Bahamas Vacations
There's a few other options - mostly condo-style resorts about a mile down the beach (or less) from Atlantis also. Some seemed reasonable for beachfront properties - you can find them on Expedia. Here's one:
Bahamas Resort - Nassau - Paradise Island Beach Club Looks to be a little older?
Bahama Divers is only a mile or so away across the bridge so a cab could work. Both operators run shuttles between Cable Beach, downtown, Paradise Island and their facilities. IDK if either pick up in that area or you'd have to go over to Atlantis. There's also local shuttles that run between Paradise Island and Cable Beach/Downtown - IDK the routes or the schedules.
Some properties on Cable Beach are priced reasonably on Expedia at certain times. Coco Plum is a real basic older option - I believe about a block or two from the water.
I've read about Stuart Cove and Bahama Divers, but it wasn't clear if they were the ones most recommended by the board.
They're the only two options unless you do a private charter with the two private companies at around $800/day.
On Grand Bahama there's the Wyndham Fortuna Beach. It's an AI resort and last time I checked had their own dive operator on-site. I don't really know anything else about it. Fred Riger at Grand Bahama Scuba also gets good reviews.
Grand Bahamas Scuba - Meet Fred Riger
Some others are in the Port Lucaya area since that's generally where they keep their boats. Bell Channel Inn is an affordable option there - the owner also has a dive boat/op at the same location. UNEXSO is another option there. They do dolphin dives there as well as shark feed dives. IDK the experience required for either. It's a yacht harbor though so beach options nearby??
The other Cove's location is at Old Bahama Bay Resort - as far west as you can get and still be on Freeport. It would be a 45min. or so drive back from there for anything else. The big diving draw there is going out to Tiger Beach on their day trips but that's something that a new diver - or a returning to diving diver - shouldn't really do. It's diving with 20' Tiger Sharks in
20' of water and they like to "bump" you. OBBR isn't exactly cheap either.
Small Hope Bay Lodge has been on my go-to list for years. All my info is somewhat older also, I haven't been to the Bahamas in a while.