There are a zillion ways to do it depending on your budget, the length of stay, what you're looking for from the holiday and whether you will hire a car or not.
If you want a beach holiday, you are looking away from Oralndo/Disney, which isnt close to a beach (closest is probably Daytona, which is nice but I wouldnt want to spend an entire holiday there). We have stayed in Anna-Maria, Bradenton, Sarasota, St Pete's etc, which all have lovely beaches (although the sea can be quite rough). You are only an hour or so's drive from Tampa airport - BA fly direct to Tampa - and a few hours drive from Orlando if you wanted to do Disney/Orlando (we always do a two-centre, but our children are slightly older and love Disney).
Have you considered Miami? If you get an apartment close to South Beach it is a lovely, relaxing holiday, although it isnt the cheapest place to eat out. It isn't a 'family' place, iyswim, which I personally like, but depends on your kids and what they like to do.
If you want a really different, chilled out holiday, you could fly into Atlanta and drive down for an hour to the Florida pan handle. There are some stunning, rugged, white sand beaches with beautiful holiday homes right on the beach up there and amazing national park land all around. Not touristy at all - its where people from Georgia/Florida holiday.