While Bimini is the next set of islands over from Florida and only about 50 miles over, besides private boat, there is no easy way of getting there. The only air service is the Chaulks Seaplane service (or private charter). (Maybe there are some small charters flying scheduled runs into south bimini, which has a paved airfield.
Even private boat, etc, you have to cross the gulf stream which can be nice and calm or have 10-12 foot seas. The Bahamas also charges $300 customs fee on private boats entering, so that hinders people going along with the $3/gallon plus gas prices.
It's just not an easy side trip from the keys without advance planning, etc.