America West has been no problem for me.
Firearms and ammunition must be in CHECKED baggage NOT in your carry-ons. The gun must be UNLOADED and in a HARD-SIDED case. That case need not be your outer luggage. For handguns, I use a small plastic case ($4.95 at gun show) locked with two tiny combo locks ($1.99 each) which goes inside my soft-side suitcase with my other stuff. You must "DECLARE" that you have the firearm. I do so by asking for "an orange firearms tag." You sign and date the tag, the agent initials it, and the orange tag goes INSIDE the plastic gun case (not outside where it might draw attention). The case is LOCKED and goes in your luggage which need not be locked.
They may or may not ask to see the gun. I don't like waving guns around in the airport, so I merely turn the case around and tell them they are free to check it over. None have.
Ammunition cannot be LOOSE. It may be in anything suitable as an ammunition storage device - manufacturer's or aftermarket cardbox, plastic ammo box, magazine, etc. The FAA rule is ambiguous as to whether boxed ammunition can be in the same luggage as the locked gun case (it says it can't be in that case). Avoid the issue and put the ammo in your other piece of your checked luggage or in your spouse's. I've never had anyone ask to look at the ammo.
The words in capitals are what the FAA requires. An airline can be more restrictive. The two times I got uninformed, difficult ticket agents, I just said "I'm not going to argue about this, call your Supervisor as I'm sure they know the rules." Neither took me up on that request and off I flew. One can be polite and firm at the same time.
Alaska Airlines has an excellent discussion on their web page. As I recall, Delta does too. If you follow those rules, you should be OK as all US airlines treat firearms in checked baggage on domestic flights pretty much the same.
Stay away from Sun Country, though.