Rich H has given you some very good suggestions. So far no one has invited you on their boat, which is what you want. I have done many Gulf Stream crossings, but have no plans on going soon, so can't offer a ride.
I have sailed from Dinner Key Marina to Bimini, from Key Largo to Cat/Gun Cay and to West End and to Nassau and to Abacos and Exumas, etc.
The Gulf Stream is nothing to fool with. Watching weather windows is important. For winter, it is best to go in later Feb, March or April than earlier because of all the cold fronts and unpredictable Northerly weather that can pop up. I have never crossed the Gulf Stream in flat conditions, but I know of some who have. I have had some pleasant crossings and some very unpleasant crossings. I'll take the pleasant ones anyday.
To go from Miami to Bimini is not a hard trip if the weather and seas are on your side. It's only like 50 miles. In Biscayne Bay there is Dinner Key Marine (Coconut Grove) or you can stay at Bayside or the Miami Marina and have easy access to Governor Cut or through the Rickenbacher and out the Cape Florida Channel. Bayside is really cool and protected from weather and Miami undesireables with lots of shops and restaurants. Some will go to No Name Harbor at the south end of Biscayne Bay and wait until midnight and then cross in a flotilla to reach Bimini or Cat Cay at daylight. I have left Miami in the late afternoon to arrive in Bimini in the middle of the night, but then you have to wait outside and anchor for customs to open up. Anchoring outside Bimini is ok if the seas are ok. Sometimes it is only recommended for day anchorages because of the sea surge. There is also Honeymoon Harbor, a relatively protected cove to anchor in as well, near Bimini. Or you can cut in between Gun Cay and Cat Cay and anchor on the east side of these islands for protection. Fly a quarantine flag and obey laws of customs. You can clear at Cat Cay, which is a private Island or Bimini. There are some stores for provisioning on Cat and you can walk around only the part of the island where the residences are not. Bimini is building up more. There is a new marina on the outside just finishing up the last time I was there. Then on the inside there are some other marinas and anchoring spots at Alice Town. Browns marina at one time was a place to go, but has since been wrecked with hurricanes. Not sure if they are rebuilding it or not. The Compleate Angler used to be there. That was Hemmingways house, but it burned unfortunately. There is the Bimini Big Game Club where the sport fish boats hang. That is fancy.
My first Gulf Stream crossing was on a 34' sailboat with only a compass. The light didn't work so had to use a flashlight. This was before GPS. RDF was around, but didn't have one. Four of us were on board and it was a very rough crossing with seas coming in from behind (better than on the nose). I was looking up at seas. Heard a mayday over the VHF, but didn't know where the person overboard was. Saw no other lights. Thought I was alone. The other three got sick and so I sailed it myself and awoke them outside of Bimini at 2:30 a.m. to help me anchor the vessel. Read about the man (actually woman) overboard in Sail Magazine some months later and it was the same mayday that I heard on the air. I was pretty busy alone out there anyway. She was rescued, thankfully.
The Gulf Stream gets faster toward the middle and not so fast at the edges. On that trip I drew out vectors on a sheet of paper to determine my best guess for a heading and when I saw the Bimini "B" light flashing I felt pretty good that I figured it out correctly. I know of some people who left Miami for Bimini and ended up back on the Florida shore farther up (like West Palm Beach). The Gulf Stream played a trick on them and they were pretty embarrassed to find them selves back in Florida after so many hours out to sea. That was long before GPS. Now with GPS it is a piece of cake as long as you have power to run them.
I think after you do your first gulf stream crossing you might be compelled to sail further into the Bahamas. They are beautiful waters and the best fun I have had almost anywhere that I have sailed (and that's a lot of places). It is a sportsman paradise for fishing, snorkeling, diving, and sailing. Go for it before you die.
