Good question. We are planning a trip to Utah this summer so I went ahead and got my out of state Utah permit. I've been searching on the net trying to learn all I can about flying with a firearm. Would love to hear others thoughts on the subject. By the way this will be my first time on an airplane as well.
Air travel with a handgun is easy. Just have it, unloaded, in a locked box in your checked bag. When you check your bag, tell the person that you have an unloaded firearm. They will ask to see it and sometimes tape a note to the box. You will fill out a card and a copy wll go inside the locked box with the gun. That's it.
Ammo should be in the factory box or plastic ammo box and does not need to be in a locked box.