There are three ways that I am aware of, as a private seller to sell your guns to someone else correctly. Keep in mind that correctly is very jurisdiction-driven, and in some ways dependent on opinion.
Make sure you can sell your gun at all. Many collectible guns fall under certain rules where you are allowed to own them if you did before such and such a date, but your no longer allowed to sell them. Also be aware that there may be county and other requirements. For example, in Florida you can sell your gun with no background check, but if the buyer doesn't have a CCL then there is still a wait period. Some counties do require background checks even if the state does not.
"Good" ways to sell
While there are exceptions these ways are so easy and accessible that most people just go one of these ways.
Gun show. There are many gun shows. If you wish to sell at a gun show there are sections set up to let you do just that. You pay a small fee, and a FFL will handle the transaction for you, like a broker.
Gun shop/clubs. Most gun shops and clubs offer a "transfer" or "broker" service. Like the gun show, a FFL brokers the deal for you for a small fee. In both cases they make sure all laws are followed and that all the paperwork is in order. They will also have rules on who the buyer can be so you can feel safe that your gun isn't going to someone unsafe.
County Sheriff's office will usually help you transact a sale if you call ahead and ask them to. They may not want to help you sell 200 guns, but they are usually more then willing to help with a few. They can do background checks, witness the sale, and make sure the paper work recommended is filled out correctly. Make sure you call to arrange this first. Showing up at the sheriffs office with a gun in your hand will make for a very bad day.
FL specific things
There is no registration in FL. In theory you could sell your gun to anyone, so long as:
- You traded something of equal value
- The buyer is a FL resident
- You know the buyer is over 18
- You know the buyer is not a convicted felon
But keeping a good record is recommended.