My fiance and I just moved to Tampa from TN a year ago so we've been playing tourist the whole time. The guys have already covered most of the spots that are on the top of our list for the Tampa / St. Pete area. But I'll include our whole list with some notes so there will be some repeats.
Bern's Steakhouse - Eclectic atmosphere but fantastic steaks and deserts as well as like 2.5 million bottles in the wine cellar. Make a reservation or you won't get in.
Noble Crust - Kind of an upscale pizza joint with great service and beer selection
The Moon Under Water - One of many good eats in downtown St. Pete but the Brit pub theme kinda stands out
John's Pass - Touristy and tons of shops and restaurants but some gems in there if you look
Honeymoon Island - State Park with great beaches
Fort DeSoto - County park with great beaches . . . they have a dag section where we bring our furry herd
Lowry Park Zoo - Pretty damn nice little zoo
Cigar City Brewing Co. - One of a ton of local brew pubs
The Tampa Riverwalk - Tons of restaurants, pubs, museums, etc.
Ulele - Upscale place close to downtown Tampa. Make a reservation if you want to get in.
SS American Victory - Restored Victory ship just outside downtown Tampa
The Florida Aquarium - Right next to the SS American Victory
Big Storm Brewing Company - Lots of great in house brews
We have a ton of other places on our list that we simply haven't had time to visit. The local brew scene is pretty big here. You can find them all over the area. Be sure to check some out. We've been to St. Pete Beach a couple times and really liked downtown St. Pete and downtown Tampa's restaurant and bar selections better. YMMV
PM me and maybe we can link up to play tour guide.