Tampa/St. Pete's for hounders with kids
I agree that Frenchy's is a good pick for families. Closer to St. Pete Beach, I'd check out Caddy's On the Beach in Treasure Island. It's a beach bar where you can eat at picnic benches on a gorgeous stretch of Sunset Beach and people watch to your heart's content while your kids frolic in the sand. The Reuben sandwich is the way to go there. Just be aware that the wait staff are way laid back and do things on their own time. . . .
When I had friends with young kids in town last summer, the kids really enjoyed some of the local Mom and Pop places I took them. Pierogi Grill is an excellent Polish restaurant in Clearwater that the kids loved. Mirage, a Persian restaurant in Clearwater with an awesome buffet and an a la carte menu, is a hoot on Fridays and Saturdays when belly dancers perform and even pull kids (and adults) on stage to join the fun. There are also a ton of kid-friendly activities on Clearwater Beach, from Captain Memo's Pirate ship to the Marine Aquarium where they can meet the famous dolphin Winter (search the St. Pete Times website for a series of articles about her amazing story), and of course the daily Sunset Festival on Pier 60 with street performers, music, and vendors.
The kids were fascinated by the sponge docks in Tarpon Springs and loved watching the waitress flame the saganaki (even if the taste of this flaming Greek cheese was a bit strong for them) at my favorite Greek restaurant, Mykonos. (If you like fresh whole fish, BTW, you may be in luck at Mykonos. Last time I was there DH and I shared a succulent hog snapper; the fisherman who'd speared it earlier in the day had just delivered it and was eating lunch there when we arrived.)
Also, make sure to get an ice cream cone at Scoops on St Pete Beach. It's owned by a couple who grew bored of retirement and opened the shop for fun. It's homey and friendly, plastered with photos of guests slurping ice cream.
And while we're talking ice cream, if you are in Dunedin or traveling up Alt 19 around Palm Harbor, get a cone at Strachen's--easily the best ice cream in the county, IMO. I'm also addicted to their carrot cake. Their original location is at Alt 19 and Tampa Road, and they have a new store on Main St in Dunedin, close to the Pinellas Trail. Also in Dunedin is Casa Tina, a wonderful Mexican restaurant with a whimsical decor. On Saturdays they clear out a couple tables for a Cirque du Soleil-style performance.
Those are a few suggestions of places that are kid-friendly, have great food, are fun, and give you a little local flavor as well.