Parker, you can hunt with confidence in Florida with a .243. The first rifle I hunted with West of Melbourne was a .243. 250 yard shot on a 200 pound hog with standard Remington Core Lokts proved medicine enough for a clean kill with a round through the lungs.
When I bought my first rifle, I got a .308 because the deal was too good to pass up. I will tell you the recoil in a bolt action .308 with 150 grain bullets is noticably stiffer than .243 with 100 grain bullets.
When it was time to buy my oldest her first rifle, I went with a .243. She shot her first deer and hog with it, using Winchester Ballistic Silvertips. The deer was @ 85 yards and she pulled the shot and spined him right above the vitals. The deer dropped in his tracks. The pig was a 250 pound sow, shot at 30 yards behind the shoulder using the same bullet (type that is, no we didn't reload the used one
). The sow stayed on her feet for 10 seconds and flopped over.
After getting her an AR in 6.8 SPC, which my oldest likes to shoot better, I find myself shooting her .243 because it is so fun to shoot! To me, there is little felt recoil.
Good luck with your purchase! There are plenty of good calibers, light in recoil, that will do the job. 7mm-08, as mentioned, is one. .257 Roberts, 6.8 SPC, 6.5 Grendel, .250 Savage, .308, .300 Savage, .260, are all short action rounds, lighter in recoil, who will all do a suitable job. Some are more easy to find than others, but with internet buying these days, you can find ammo for any online!