Alice Town has a small museum which is interesting. There are some restaurnants and a lot of bars. You can take the water taxi to South Bimini and look for the Fountain of Youth that is supposed to be over there. You can walk down to the point where there is a cemetery. You can buy Bimini bread and coconut candy or T-shirts at the Straw Market.
Other than that, fishing, diving, swiming with wild dolphins or snorkeling is the thing to do in Bimini.
That and relaxing.
Only 2 things to do on Bimini. Fish or Dive. We dove. We used Keefe's for diving. Good operation with great people. First day there was about 18 people diving, but for the remainder of the week there was only 6 of us. Nice to have the ocean to ourselves not to mention all the room on the boat. We were propositioned by several locals at the dock for fishing trips but did not have the time to do so - sorry I had to miss the fishing. There are several restaurants on the main drag that will cook your catch for you.