Most airlines policies are in line with the TSA rules, but it's always best to check each airlines individual websites.
https://www.tsa.gov/travel/transporting-firearms-and-ammunition.
Remember, when travelling with firearms, you will always be at the mercy of the individuals on duty at the desk at that time. Most TSA and airline personnel are not gun people and although I have never had any problems getting on a plane with everything I brought, it's been a close call a few times. Be polite and professional, and carry copies of TSA and the airlines specific policies with you. Never argue with a gate agent, simply ask for a supervisor to be brought out to clear it up.
When it comes to ammo, If you want to be 100% certain you're good, have the ammo in original packaging where every round is housed in it's own separate space. Have the ammo inside a locked hardcase just like the firearms, and don't carry more than 11lbs.
Some gotchas:
Loose bulk packs are "original packaging", but may not meet that individual gate agents smell test... a 500ct bulk pack of loose .22lr may be rejected by some airline employees because each round is not separately contained... but all your normal 20ct rifle and 50ct pistol boxes have each round individually stored. That's the way an airline employee expects it to look and won't bat an eye. For handloaders, normal plastic Midway or Berry's boxes are also fine. There is no requirement by the TSA to have your ammunition also inside a locked hardcase... it can be carried just stashed in your checked luggage with your undies. Airline employees might not like that either, or assume the ammo needs to be inside a locked case like firearms.
The TSA specifically says that loaded magazines are fine, and that it's also fine to have those loaded magazines in the same locked case as your firearm. Although magazines must be inside a locked hardcase, and can be in the same case as your guns, a couple airlines specifically prohibit loaded magazines. To my knowledge, NO airline prohibits you from carrying ammo in the same case as your firearm. Again, some airline employees might assume you CANNOT transport a loaded mag, or that you cannot put mags inside the locked hardcase, or that whatever else they misheard or wasn't taught to them. Take copies of their policies.
11lb max limit for ammo weight. Airline employees have no idea how much ammo weighs, and a quick Google search reveals nothing I could find of anybody saying they were required to offload their ammo for a separate weigh-test to ensure they're within compliance. 11lbs of .223 is 465rds. I know that I've transported more than 11lbs on at least a few trips. Spread the ammo over a couple bags if you think you're close.
When I travel with firearms, I always err to the side of caution and don't set myself up for a misunderstanding about the rules.
All my ammo is in factory or plastic cases, no bulk packs. No loaded magazines. My long guns are individually trigger-locked or cable-locked inside a Pelican 1720 that is also locked. I will usually put a pistol inside a smaller Pelican case inside my checked luggage and not in the larger rifle case just in case the rifle case doesn't make it, I still have my pistol wherever I land. I always use TSA locks on my checked bags. I always book extra time to hang out down at the ticket counter if my bags are going to be scanned behind the wall. In some airports, you take your bags directly to TSA scanners in the ticketing area and can watch them go through the machine. If your bags are going directly behind the counter and will be scanned out of your sight, there is always the risk that they'll find something they don't like, and you'll be required to come BACK to the counter... which sucks if you head directly to your gate a mile away. I hang out 15 minutes near the ticket counter just in case.
Lastly... remember that neither TSA no airline employees have any training on "gun stuff". They are only trained on the specific policies that they are required to know. They will naturally be overly cautious with anything to do with guns and will always take the most conservative line when it comes to policies. Set yourself up for success and you'll have no problems.