While I'm a big WDW fan, my absolute favorite spot in Florida is Fort Myers! I have vacationed there for 1-2 weeks per year for about 15 years, usually staying down on Ft. Myers Beach. It is a very beautiful city, which I've explored numerous times. The beach is wonderful, the marina downtown is beautiful in a different way, there's 2 baseball teams that have spring training in Ft. Myers, there's history (Henry Ford and Thomas Edison winter homes - a very interesting tour), and of course plenty of good restaurants and shopping. Some parts get really congested at the height of tourist season, especially Ft. Myers Beach, but that's to be expected.
The airport is really nice too. It's small enough that it's not wild like some of the big airports, but yet it has plenty of airlines and flights coming and going. They've expanded the airport several times over the years.
One other good thing is that Ft. Myers is very close to Sanibel Island. You will LOVE Sanibel with all its tropical growth. Other nice communities nearby are Bonita Springs and then the upscale areas of Naples and Marco Island.
Being that far south in Florida gives you a better chance of nice warm weather for much of the winter months. Of course, it may be more beastly hot in the summer. Not sure, because I've only been to Fort Myers between December and May.
Hope that helps a little. If you have any specific questions, I can TRY to answer. I'm envious of your move!

Good luck to you and your family as you relocate to a beautiful city!
