Ed, my brother in law is from Tampa, and my sister lived there for a while. I used to head down to visit every year in March when New England winters begin to lose their charm. (They've moved to Chicago now--it's not quite the same.) Even without local knowledge, I found pretty fair fishing for reds and trout poking around with the kayak. There was an area around Fort Desoto that was quite good, and then a big mangrove complex south of Tampa that was a "no motors" refuge of some sort. I don't know if hunting is allowed in there, but fishing is, and I got my first tailing redfish in there. I can't remember the name of the place, unfortunately.
We also launched a couple of places off the Sunshine Skyway and found trout.
If you want to tempt the gators, take that kayak south a bit and paddle Myakka River from the state park down towards tidewater.
Just stay away from the tourists feeding the gators marshmallows.