The key to the whole thing lies in the word concealed. Florida does not recognize open carry. Even with a permit to carry a gun, the gun has to be concealed from sight. Hence; Concealed carry permit. Having a permit to carry does not mean that you can carry your gun everywhere. You cannot. There are definitive laws with regard to this. It is serious business.
Florida's Department of Agriculture claims to have dramatically decreased wait time for a permit so you should receive your permit pretty quickly if there are no issues with your record or fingerprints. Sometimes, fingerprints (manual or electronic) aren't adequate, for whatever reason, and have to be re-done. I seem to remember that they will take them twice and if they still are not able to be used, a manual check is done by the FBI. Turnaround time is still pretty fast. Their website is good about posting your application status.
If you have a carry permit in Florida, you can buy a gun without having to endure the 3-day waiting period but you are still subjected to and pay a minimal fee for, a background check. If it comes back clear, you are free to take the gun with you.
Good luck and stay safe.