I've flown several times with firearms. All you need to do is to:
1.Give yourself a little extra time. 15 minutes depending on how busy TSA agents are. You will only have to check your firearm at the beginning of your flight and at the beginning of the return if roundtrip. No connections just like any other checked luggage.
2. Have your firearm in a hard sided case. I recommend a Pelican case, aluminum cases will do, but they will come back with some nasty dents and dings.
3. The case must have the ability to lock. Built in locks are OK. Pelican makes locks that have a key ability that TSA posseses according to Pelican.
4. Don't get too close when they are verifying the firearm is unloaded. You will sign the red form stating firearm is safe and unloaded. And they will check it themselves to make sure it's unloaded.
It's been fairly easy at Colorado Springs, CO; Greenville/Spartanburg, SC; Charlotte, NC; Elmira/Corning, NY; Norfolk, VA.
BTW your ammo can only be in a factory ammo box, or a box designed for ammunition inside the case. One time I left 4 rounds that were in the sling and they were confiscated.