To take a gun in checked luggage, you need to call the specific airline and ask about any additional regulations. But basically, you need to have the gun in a clclosed (not locked) box within a piece of luggage (closed, not locked). You need to tell the person at the counter that you have an unloaded gun in there. It is best not to bring any ammo, but certainly none may be in the suitcase. they will make you sign a form that the gun is unloaded etc. You should also have a trigger lock on it. If it is a revolver, swing the cylinder out, and put either a padlock or cable lock through the frame so that the gun is inoperable.
You should then have no problem with the airlines.
However, a problem on the ground still can occur, "Carrying" a gun is ill defined, and even in states where possesion of a gun is ok, "carrying" may be a problem. that is why it is best to have no ammo ony you, and have the gun locked, and double, and triple locked. then you are only transporting. In some states, the state patrol will tell you that that is also illegal.
One mitigating factor is if you are on your way to an NRA-sposored shooting event (and have some event info to prove it).
Generally, if you are flying from Memphis to Birmingham, you will have few problems. New York to Boston could in theory, be problematical.
In upstate New York, it is (relatively) easy to get a gun license. However, that license is NOT valid in NYC. So if you fly out of LaGuardia or JFK, there could be problems.
Obviously, it all depends first and foremost on the state. I believe that CA has very restrictive gun laws. (What a surprise, Barbie Boxer and DiFei for senators.) Check specifics like the NRA website for details about the state law. Tread carefully here, if you violate these laws, it is usually a felony.