Welcome to the forum Russell.
Your not in a bad area to get into some hunting.
Nearby you have the public land called Green Swamphttp://myfwc.com/hunting/wma-brochures/sw/green-swamp/
Season is still open, but you have to read the regulations as to dates and such ( I believe only for a couple more weeks)
I could probably type for hours and never give you all the info you could use for hunting Florida, but I will give you some starting points.
FL public lands are called WMA's (WIldlife Management Areas). Your area is Southwest (although you are not limited to that zone, you can hunt anywhere, I am just going to focus on that)http://myfwc.com/hunting/wma-brochures#wma
Florida has first come first serve permit WMA's (Like the Green Swamp**), you find the place you want, check the dates, get there early and you hunt. Deer, small game, hog seasons etc. (hog is almost a year round staple).
They then have limited entry/quota hunt WMA's like Upper Hillsborough (near you). http://myfwc.com/license/limited-entry-hunts/
These hunts you have to apply early in the season and hope your name gets drawn. You register for whatever hunt you would like (start here https://license.myfwc.com/
) for the upcoming season. There is like a 30 day window to apply and I don't recall the exact start dates offhand. YOu register, if you get drawn they let you know (if you dont get a draw, you get a preference point for the following year). Often times, the have 3 phases as well on these draws, so if you miss phase 1 sign up for phase 2 (watch the sign up dates). If you are awarded any of these permits they are all free.
There are also special Opp hunts. These often require an extra fee. One I like near you is Green Swamp WEST
. Its $5 per entry, you can enter as many times as you like, and if selected you pay another fee (GSW is $100). There are a few like this around the state.
We also have National Wildlife Refuges that get opened to hunting (some are draws, some are walkups) like Lower Suwanee, St Marks, St Vincents (get a chance to kill a Sambar deer!). All this info is on FWC as well.
I have mostly geared this towards deer hunting, however we have the same type of process that you would go through for Turkey, small game, hog, gator, waterfowl etc. (hog can be killed on just about any WMA during deer and small game season (gives you almost 6 mos in your area)). There are similar links on the FWC page for these seasons.
On the license page you would need a FL hunting license, WMA stamp and deer stamp. Archery stamp too if you go that route. You can buy a Sportsman license as well if your going to utilize all that.
As for overall hunting, its like any other public land, perhaps a tad more effort to get on the animals. Go where no one else wants to go or can go. Deep, thick, wet swampy nasty stuff a 1/2 mile + into the woods/swamp if you want regular success.
We do have liberal seasons and bag limits as a plus. Your current zone (B) deer season started in Oct and goes till Feb 23.
Check out the FWC site, let me know if you have questions I can try to help
** Some of the first come first serve have the first 9 days blocked out IE you have apply and get drawn, then after that its FCFS)