If you're looking for local places, the closest gun shop is Take Aim in Palm Harbor. Located on the East side of US 19 near the Innisbrook entrance. Won't be a memorable experience, but it's a good shop.
Another close place is Coles in Dunedin. I don't remember the exact address but I think it's something like the corner of Sunset Pt and Douglas Ave. A very small shop, but interesting stuff from modern to late 1800s. I like to paw through their C&R stuff.
Others have already mentioned some good places in the vicinity.
If you're into "old stuff" (pronounced C&R), you gotta check out Mark Kubes place off of Ulmerton in Largo. Lots of Mosins and Mausers.
But then, if you're into old stuff, I HIGHLY recommend the Lakeland show this weekend. Wall to wall C&R and modern stuff. From Palm Harbor it's probably a 45 minute drive.
LL