Let me add a few extras to this list.It's rather simple. You put the gun/guns in a hard-sided case that is able to be locked. You bring the case to the counter at the airport and tell them you need to declare a firearm. You MAY have to show that it is unloaded and they will have you fill out a little declaration card stating that it is unloaded. That card will go in the case and then they will have a TSA agent inspect the case (in your presence). After they inspect it, they will have YOU lock it and you will retain the keys to the locks. It will then get put in with the regular baggage and will be checked through to your location.
Upon arrival at your destination, you will pick up your gun with your regular checked baggage in most cases. In some cases they pull it aside and page you to the baggage check office. No big deal either way.
I traveled from Chicago to Spokane last fall with 4 handguns and it was easy and painless.
Ammo can be placed in your regular checked luggage. It needs to be in a container that is designed to hold ammo so it's not loose. (factory boxes, magazines with covers, plastic ammo box)
Each airline has weight restrictions on how much ammo you can bring. Check with your airlines website as to their specific policy.