Shipping the shotgun is fairly easy, but shipping the handgun is a different story. The shotgun can be sent USPS, FedEx or UPS, with adult signature required, and no ammo can be included. Make sure there are no markings on the box indicating that it is a firearm. The handgun, on the other hand, can either be shipped via an FFL or you can ship it yourself, HOWEVER, in order to ship yourself, it MUST be shipped overnight via UPS or FedEx, and you must take the gun to one of their facilities and declare that you are shipping a handgun (you cannot take it to Staples, or a UPS Store, or Mail Boxes Etc). Handguns can only be shipped via USPS if they are shipped by an FFL - it is illegal for a non-FFL to ship a handgun through the mail.
I always ship my handguns through my FFL, but some will charge you a fee to do this, so check around.
Both Fed Ex and UPS require that either the shipper or recipient be an FFL. Although BATFE says you may ship a firearm to yourself. Fed Ex and UPS policies require the afore mentioned restrictions