Here's my take: The transition from average recreational diver to cave diver has a number of components. One, you have to switch to diving cave-acceptable equipment, whether that's backmounted doubles or a sidemount setup. Two, you have to bring your buoyancy, trim, and general control in the water up to cave standard. Three, you have to develop your situational awareness to allow you to be safe in a cave, and if you intend to dive with other people, your awareness of and communication with your team. And four, you have to learn how to run line, do tie-offs, and put in jumps, and you have to learn the marking conventions for cave navigation.
Now, if you look at the first three things on that list, none of them require you to be in a cave to do them. You can dive doubles in OW, refine your buoyancy, and train your situational awareness on any reef dive. It is my VERY strong belief that those things SHOULD be done in open water, and save the cave time for learning the things you have to have a cave to learn.
Now, your situation, being in Florida, is a little different, because you may be able to drive easily to where you can dive in a cavern. If so, a cavern class may be the place to start -- but if you have had no introduction to the standards used in cave diving, it may be a rude surprise. This is the reason I recommend classes like GUE Fundamentals (available in multiple places in Florida) as a starting point. If you come into your cavern class, and you can hover, motionless, six inches off a silty floor, and use a good frog kick, and do a good helicopter turn, and swim comfortably with a mask off, you will be WAY ahead of the game in cavern, and your instructor can spend more of your time in the overhead, working on line skills and the like.
If you have worked on your basic diving skills as enumerated above, and gotten them to cave standard, by all means do both in one trip. I did. You'll be ready.