There's really not really anywhere to do any kind of road riding down there, not saying you couldn't go for a long ride on A1A, US1, Overseas Hwy, but most people down there are tourists, and they aren't looking out for bikes. A lot of residents are seasonal too, so there might not be a strong enough road bike culture for a club ? Hell I could be completly wrong, and you may find a huge group of road bike people, and I hope you do !
Back in the late 80's I bartended for the summer right off Duval, I lived with a friend in a tiny tiny house by Beach Dawg park. Even back then we could barely afford rent, but man it was probably one of the best times in my life, at least the parts I can remember. No idea what your situation is, but Key West has gotten so expensive to live there. I'm a old school kind of guy, I was there before Margarita Ville, and when Sloppy Joes was just a cool little dirt hole off Duval. The island to me has lost all its old time charm and become way to commercialized, the cruise ships ruined it for me. It's so popular now that all the people that visit are seriously impacting the reefs and fishery. Used to be pretty bright green and turquoise colored water from Key Largo to Key West, now there's big patches of dark green to kind of brown water when driving through Marathon and Sugar Key.
Hope you get to live there for a while and enjoy the tropical weather, good luck !