Back before some priority-changing life events, I used to do a lot of firearms training ("sometimes an instructor, always as a student" as one of my late mentors used to remind us). As often as not this required travel, sometimes by car, sometimes by plane.
One of the provisions of the Firearm Owners' Protection Act is an affirmation of the right of safe passage from one state to another even if one is traveling through a state that is extremely gun-unfriendly. In a nutshell, if you're legal where you started and legal where you end up, so long as you're just passing through (normal stops for food and gas) you're legal for the whole trip so long as the guns are unloaded and not immediately accessible (locked in a box if you don't have a trunk).
Flying with guns isn't as difficult as it seems. You're just required to check them unloaded in a locked bag. In fact, some astute security-minded folks figured out that you could put a starter pistol in with your expensive camera equipment if you wanted to have a "real" lock, not one of those cheesy TSA locks which everybody and his brother can now download and 3D print master keys for. Those TSA locks arguably don't meet the requirements for locks on checked baggage with guns in it because of the aforementioned master keys.
But what's the fine print on traveling with guns and how do you avoid misunderstandings and drama? As someone who's traveled with guns a fair amount here are my rules:
When you pack the guns, lock the action open or put a chamber safety flag in the action. The bored TSA guys will want to inspect your guns to make sure they're unloaded; they will appreciate that you made it easy to visually verify that there's no ammo in the gun without handling or waving it about... assuming that they're clued in enough to be aware of muzzle control and keeping the gun pointed in a safe direction. I wouldn't take that bet though.
Magazines out of the guns and empty. Ammunition in original packaging or other boxes specifically designed for this purpose. The latter is actually a legal requirement at 49 CFR §175.10(8), and no, a ziploc baggie does not count. You can pack the ammo in the same piece of baggage as the guns.
Mind your language. You are required to "declare" the firearm and/or ammunition when you check your bag at the counter, yet if you are a guy with a short haircut like me, saying "I would like to declare a gun" is as likely as not to get you presented with the paperwork for law enforcement flying armed. This isn't what you want. Exclaiming "I have a gun!" is of course Right Out. My preferred phrase is "I need one of those orange (or white, depending on the airline) 'Unloaded Firearms' tags, please". Depending on where you're flying from, confusion and calling a supervisor may result (DCA), or they may point to a stack of them on the counter next to the luggage nametags and tell you to help yourself (JAC) or something in between, but regardless you've told them exactly what you need in a way that could not possibly be misconstrued. The tag goes inside your locked bag, not on the outside as a big conspicuous "steal me" flag. This is also a legal requirement, see 18 USC §922(e).
Avoid places that are culturally hostile. Notwithstanding the protections afforded by the Firearms Owners' Protection Act, you can expect to be detained "in good faith" by law enforcement until cooler heads who actually know the law prevail if you attempt to check firearms in a place where extreme hostility to firearm owners is the norm. Places like Chicago and any airport run by the Port Authority of New York & New Jersey come immediately to mind. Oddly enough I've done OK at SJC and BWI.
Know your airline's restrictions. Rifle cases can be too long. Any gun case can be too heavy (if you've got a big pile of guns). Most airlines restrict by weight how much ammunition you can bring with you - enough for a weekend class at one of the famous shooting schools is almost certainly way over the limit, so plan ahead, explore opportunities for drop shipping the ammuntion to your destination, etc.