The tl:dr version: For the classic beach-time vacation check out Treasure Island (between St Pete and Clearwater). That's become our annual family trip for our family of four. (kids aged 9 and 11).
Now the detailed version I originally wrote... I've been visiting Tampa in the winter to see family for 10 years. It started myself but this year became trip #4 with the lady friend and kids. We've been transitioning more into our family vacation than just visiting my parents. We stayed in Treasure Island 2 years ago and loved it. We actually bought a timeshare week at the place we stayed and go there annually now. Its mostly low rise hotels and condos in those middle islands between St Pete and Clearwater. Its still an island chain so Gulf Boulevard is the only thouroughfare north and south, but the area has a fraction of the congestion and population density as other areas. Our place borders the beach and its our home away from home for the week. We drive into the mainland after unpacking (the 10 minute drive is worth it, the islands all have Publix stores but they're much smaller and super expensive), load up on groceries and supplies at Publix, Winn Dixie, or Walmart and set up camp for the week. If you're not looking for the theme park type deal and just want the beach vacation thing then its great. There are a lot of smaller activities that you can do nearby, and plenty of shopping centers not far from you on the mainland. The lady friend does much of the annual shopping for the kids down there and brings bags of clothes home. Johns pass is a boardwalk village in Madeira beach that does tons of boat\jet ski rentals, dolphin and sightseeing cruises, fishing charters, bars and restaurants and a host more of other things. We go there several times during the week just for something to do to take a break from beach. There's a boutique winery in there as well as a hand rolled cigar shop I get supplies from for after sundown when we retreat from the beach back to the pool and hot tub.
You're generally not more than an hour away if you do want to go to a destination thing like zoos or aquariums. You can also shoot a half hour south to Fort Dr Soto park. Its an old army fort with tons of beaches and picnic areas. I've never done it but they rent kayaks and paddleboards, bikes, and there's also a ferry service that goes to a couple islands only accessible by boat.
The area has practically zero branded hotel chains, its just a blend of different places. Therefore Google and the vacation search sites don't show the bulk of available lodging. Its mostly mom and pop places and airbnbs. A lot of places you actually have to know about and seek them out to find out about weekly rentals because they won't show up in search engines. It blows my mind how these places have websites but are still so underground. Many are condo communities that have unsold units or owners putting weeks into the rental pool so you have to actually call and inquire. If youre interested (or anyone else for that matter) I can tell you a few places like that around Treasure Island. Its a lot like that all the way up to Indian shores but I don't know many of those places.
In summary, our kids are younger, but if you're looking for the classic relaxing beach vacation then I highly recommend the area. Our days start and end at the beach, and each afternoon there's something nearby to make an excursion if you choose.