.... In our current world I can't believe that it is as easy as just getting in the boat and going.
Of course, it isn't... As Donna alluded to you can't simply buy a boat head out of Miami harbour and magically 'live the dream life'... - but the biggest issues are not going to be the bureaucratic ones..
There are many things for you to sort out.
Budget is the first.. we get a lot of posters like yourself that want to do this sort of thing on $20K - not realistic. A decent, appropriate boat for what you plan to do is likely going to run around $100K or more, especially once you've upgraded, fixed and prepared for such a voyage. Even if money's no object and you spend half a mill on a superb boat it's going to take you months or more to sort out all the systems and learn how to handle her - esp with no significant sailing experience to start with. Much like flying, it's the landings and takeoffs that can be most challenging, esp with bigger boats.
"The plan" is another... Buy a boat in the NE and sail/traverse the ICW, then the 'thorny path' to get to the "idyllic islands"? Or buy a boat already there? Do it next week? or with a reasonable couple of years to prepare yourselves and the boat? All these decisions are going to make a big difference to your ultimate success, as will your own personal fortitude and skillsets. Each chosen 'path' has it's own challenges and issues.
Have you every experienced ocean swells? Storm conditions at sea? It's not necessary to have survived the 'perfect storm', but being physically and mentally prepared for the eventuality is important.
Can you get some sailing experience on the Lakes in the meantime? Can you spend some time and money chartering boats similar to what you'd like to get first hand the 'feel' and handling of such a boat? Other than the crowding I believe a charter in the Virgins may be a good way to get a taste. No long passages, relative shelter. A skippered charter would make most sense for the first time. It's at some cost but that may well pay off down the road.
Anyhow, best of luck. It's an exciting idea, and is doable as long as you stick to the sentiment in my signature