My understanding is that unless otherwise marked, you can dive in public waterways. Now, it's not always safe to do that.
I've often thought of diving clearwater beach myself - it's about 15 minutes from my house. There is supposedly nothing to see there, but I was planning to bring a metal detector and go hunting for gold (a friend of mine lost his wedding band there, I bet it's common). Hillsborough river is kind of nasty. I'd bring a drysuit and FFM personally, but I'm sure some people do it with wetsuits or less.
In some places it's prohibited to scuba dive in a marked swimming area (such as Blue Heron Bridge). I believe you can be given a ticket and small fine. I'm not sure if that rule applies to all marked swimming areas - if so, it would put Clearwater beach off limits. Also, you're required to have a dive flag, or you can get at ticket (or run over by a boat).
If you're taking a dive class in Florida, they will probably touch on this some in class. If they don't, then ask the instructor before class is over.
Welcome to diving, and to scubaboard!
P.S. I like to use this site to find divesites:
Scuba Earth by DiveBuddy.com | DiveBuddy.com I'm not affiliated with the site or anything like that, it's just been useful for me.