You should fly a quarantine flag when you first enter a Bahamian port showing that you have not cleared yet. If you arrive in the middle of the night and anchor out until the customs office opens, then you'd better be flying your yellow flag and don't fish or get off the boat until you are cleared. When you go into Alice Town (say Brown's Marina) everyone stays on the boat and the skipper takes the passports or crew's ID info, plus boat papers and goes to clear customs and immigration. After that is completed, the skipper returns to the boat and everyone is ok to get off the boat and enjoy the area. The skipper will get a cruising pass. If you want to take lobster (if the season is open) with a Hawaiian Sling, you must request that they write that in on the cruising permit. It is $150 for a boat 35 feet or under. $300 above. Another place to clear customs is Cat Key, which is south of Bimini. Go through the cut where the Gun Cay light is and it is a short motor to the Cat Cay Marina. The island is private, but you can walk around much of it (the rest is off limits) and there is some provisions there. Honey moon harbour is a place to anchor if you are looking for a place to go. Along the western shore of Bimini you can anchor during the day, but take caution during the night due to swells. Inside the Gun Cay light there is anchorage and south around Dollar Harbour is some good anchorage, but sometimes the current runs swift there so be careful.
Of course you can anchor inside the cut into Bimini. Brown's Marina has been rebuilt after Hurricane Andrew. It was a heap pile many years. I have stayed at Brown's before it was demolished. The Compleate Angler, Hemmingway's house unfortunately burned down a while back, so that is no longer a destination. Bimini is building up like no tomorrow now, though. Other info above informs you of re-entering the U.S. and make sure you get your decal before you leave. You want to have that anytime you are offshore (even if you haven't visited a foreign country).