I've lived most of my life in or near Tampa, and it's actually a pretty decent family destination. Not as many things to do as Orlando, but then almost no place outside NY/LA has as much to do as Orlando.
If you want beachfront in Tampa, Clearwater Beach was been rated as having some of the world's best sand by some egghead oceanography profesor at the University of Maryland. I really dig the vibe on Longboat Key and Anna Maria Island in Sarasota, personally.
Things to do:
Busch Gardens, Tampa
Adventure Island (huge water park), Tampa
Florida Aquarium, Tampa
Salvador Dali Museum, Clearwater
Mote Marine Aquarium and Bishop Planetarium, Bradenton
Museum of Science and Industry, Tampa (kids love all the hands-on stuff there)
And depending on when you're here, there's MLB Spring training throughout the area, the Florida State Fair is in the same County as Tampa, as well as the Plant City Strawberry Festival and Hillsborough County Fair. And the Rays play just South of Clearwater, in St. Petersburg. Consistently among the cheapest places to see a regular season MLB game. Lightning Hockey in Tampa, if you dig Hockey.
Just be aware , though, that if you stay on the Gulf Coast of FL, the waves will be smaller than on the Atlantic Coast. Depending on how small the kids are, that may be a plus or a minus.
Down South of here, Marco Island and Naples have fantastic beaches, but I don't know what hotel prices are like down there. Real estate in general is much higher down there, so I would expect higher hotel rates. The drawback to Naples/Marco is that there's not a whole lot to do right in that area for families with kids. The upside is that Miami is 90 minutes east and Tampa is 90-120 minutes North, so you have 2 metro areas in driving distance.
On the other hand, if you stay in Clearwater, you're already in a major metro area with plenty to do, and Kissimmee (Disney) is less than 90 minutes East if you want to go there.
You can often save a few bucks on airfare by flying into St. Petersburg if possible, too. If you're flying in, that is.