I've shipped a few handguns for a variety of reasons. Anybody can ship a handgun to an FFL by common carrier. For modern handguns, having a C&R doesn't do anything for you, really.
You'll need to pack the handgun well, and take it either to UPS or FedEx. It needs to be one of their actual hub/office locations...UPS store or a drop box don't count. You must also declare it as a handgun...Being cute and putting "machine parts" or something is inviting trouble. If they find a handgun in their system that you didn't declare, you will hear from the police. Also good luck on collecting insurance for undeclared or improperly declared items.
Be prepared to take documentation proving that the ship-to address is indeed an FFL. Easiest way to do this is to ask the receiving FFL for the first 3 and last 5 digits of their FFL then go over to ATF FFLeZCheck to look them up and print out the report. It's not foolproof, though...If the address on their license doesn't match the ship-to address they gave you, UPS/FedEx will likely decline to accept the package. Also, some FFLs will refuse to give you this info for whatever reason, which will make things difficult.
Usually, if you declare it and have reasonable proof that it is going to a legit FFL, the process is pretty smooth. Occasionally, though, you'll bump into someone at the counter that doesn't know the rules/regs or just doesn't want to deal with you and they'll tell you you can't ship handguns. Sometimes if you're polite and can get a supervisor you can get it sorted, but sometimes I've just had to try again later.
UPS and FexEx both require handguns to ship via next or second day air. This is going to cost you somewhere between $50 and $100. Again, don't try to be cute and short-cut this by improperly declaring the contents. That'll bite you in the ass eventually.
So, while it can be done, it can be a pain in the ass and it's expensive. It's far easier to make friends with a local FFL and pay them to ship the handgun for you. The rules are slightly different for 01 FFLs...They can use the USPS and ship stuff common carrier ground, so it's less expensive overall to go this route than to hassle with doing it yourself IMO. If you're going to be doing this a bunch, you should consider just getting an 01 FFL yourself. That can also short-circuit any questions of you being "in the business" of selling firearms without a license...
At the end of the day, I generally just sell or trade stuff locally since the hassle of shipping isn't worth the extra money unless the item is rare.
Somebody mentioned Armslist...I'd recommend being cautious with that. It's like Craigslist for guns...Maybe the deal goes smoothly, maybe you get robbed. If you do sell stuff there, NEVER let them come to your house. Meet in the parking lot of the police station or something.