As long as you satisfy the criteria for legal ownership of a handgun and are a Florida resident, you're good to go. There's no legal requirement to complete a bill of sale. You don't have to register the gun. You don't have to notify anyone. Pay the man...take the gun...head for the range.
"An armed society is a polite society." Robert A. Heinlein
"After a shooting spree, they always want to take the guns away from the people who didn't do it. I sure as hell wouldn't want to live in a society where the only people allowed guns are the police and the military."
William S. Burroughs