Long guns can ship via the USPS (as well as UPS or FEDEX . . . FedEx ground is pretty affordable).
Hand guns MUST ship by "common carrier," such as UPS or FedEX. Both of those carriers require that you ship handguns Next or 2nd Day Air. Figure about $40 to $50 to ship most handguns. You do have to declare that you are shipping a firearm when you use these services.
When you ship a firearm to an FFL INCLUDE A COPY OF YOUR DRIVER'S LICENSE. No, a license copy is not required by law, but many FFL's will not accept a shipment without it. An FFL must only record the name and address of the person he received the gun from, but many dealers are not comfortable with just that information. They want to see an ID. Their shop, their rules . . .