Went to one of the outer/"family" islands in the Bahamas a number of years ago - Eleuthera. Slower paced than say, Nassau (New Providence island) - much less touristy. The whole island is only 110 miles long and very narrow - easy enough to explore. Good diving/snorkeling around there. Fishing too, if you like that sort of thing. Beaches are awesome (French Leave, Lighthouse, Ten Bay, Surfer's) - if you surf, definitely go to Surfer's Beach - the waves weren't immense by any stretch, when we were there, but it was fun to watch - a few tried it, and enjoyed it.
Ocean Hole might be fun, but I don't know what you're looking for, exactly - you can swim there, and feed the fish (no fishing there though). We never went, and I can't recall the name of the place, but we heard there is a plantation/farm that does tours and they used to grow a whole schwack of hot peppers that they exported and also produced their own hot sauces, etc. If you like hot sauces, Eleuthera lays claim to a number of different brands on the island, that are supposed to be pretty good.
There are some caves that can be fun to explore - Preacher's Cave (Tay Bay Beach is pretty much right there, and really beautiful, so you could explore and then spend time on the beach), Hatchet Bay, The Cathedral, Bat's Cave (Ten Bay Cave), Gale Hole, one that we were told is called Smuggler's Cave but later heard that isn't the official name of it). Glass Window was interesting (Atlantic on one side, Caribbean on the other, separated by a narrow strip of rock) - we stopped on the way back from there and bought fresh pineapple from a roadside stand. So good - they took the outside skin off and left the top tuft on, to hold, and you just eat it like that.
Some of the towns are fun to wander around in - if you're in Tarpum Bay, eat at Barbie's; stop at Tippy's in Governor's Harbour. I dunno - we found a fair amount to keep busy with when we wanted to. Blackbeard is alleged to have spent time on Eleuthera and there is a dive specific to Blackbeard's Treasure Hole (lol) - Eleuthera lays claim to a few somewhat famous pirates and allegedly, there is still some treasure belonging to Blackbeard, that is squirreled away on the island somewhere, and not yet found. We were there during the Goombay Summer Festival and enjoyed that. There are a few resorts on Eleuthera, and a variety of private homes for rent, all over the island.If I had a pile of moola to spend, I'd stay at
Five Palms Beach House.