To the best of my knowledge, to have a transport a firearm via commercial (domestic US) airline, the following are needed, AS APPLIES ONLY TO CHECKED-IN LUGGAGE!!!!
1. Firearm in a TSA-Approved lockable solid-sided (hard plastic that does not bend, or aluminum/steel, or any combination).
2. Firearm unloaded, no magazines inserted, no ammo in magazines. Ammo box may be stored in the same case BUT I have heard of instances when the number of rounds in the box became an issue....do not know if this is policy or simply overzealousness on the part of the inspector.
3. Be aware that you will most likely lose the locks, and they may be replaced with other TSA-Approved locks.
4. Be aware that transporting a firearm is OK and legal from a point of origin that allows that firearm to a point of destination that also allows that firearm. If there are stopovers in between in any city that does not allow that firearm, and the commuter (meaning "you") have to take possession of your baggage, you are taking the firearm into your possession, which can be illegal in some jurisdictions. For instance (PURELY HYPOTHETICALLY), a Barret M82A1 is legal in AZ and NV and OR and WA, but illegal in CA. If your flight is nonstop from say AZ to WA, no problem. If there is a stopover (no change plane) in CA, as long as you do not handle your baggage or take possession of it, you SHOULD be OK (a lot depends on the level of zeal of the TSA inspectors). If the stop in CA involves anything where you will put your hands on your luggage and it is not in the uninterrupted custody of the airline+airport+TSA, it is illegal import of a banned firearm. If you are taking it from AZ to AK but with a 24-hour or whatever stopover in Canada, it is illegal import into Canada of a military firearm....hunting rifles are often allowed but not the Barret .50BMG. Absolutely no firearms are allowed into Mexico via air travel unless there are extremely unique circumstances (i.e., permits secured from Mexican Gov't prior to flight).
5. To my knowledge, TSA has discontinued their practice of wrapping bright orange or yellow tape that says "CONTAINS FIREARM" on luggage.
6. When checking in at airport, go to counter of your flight, but do not fall in line. Wait for any TSA personnel to walk by, and politely say you would like to speak to a TSA Supervisor. If asked why, POLITELY request to speak only to a Supervisor. When one shows up, clearly state that you are checking into a flight on that airline and would like to check in a firearm. You will (usually) be taken to the back (behind the counter) where all your luggage will be x-rayed, then opened and physically inspected, then re-locked, and your paperwork (boarding pass) will be processed, and you are then clear to go to the passenger security screening area.
DONE.
If the question was one of bringing a firearm INTO the US from a foreign country, I would recommend contacting the TSA as well as the equivalent authorities in the country of origin. Bringing a firearm OUT of the US to a foreign country, there is not so much issue on the US side....the issue would be in the destination country and any other stopover countries (if any).
I hope that helps.
Cheers!