Ok, you won't be landing in DC, I believe. Reagan and Dulles are in VA and BWI is in MD so, it will work in your favor if you are in VA. Be very careful, and research the statutes yourself as DC has very strict possession laws on handguns, and any violation is a federal felony. MD is almost as bad if you are going in and out of BWI.
Otherwise, flying is no issue. Check in at the counter are declare your firearm. Keep it in a locked case with a non TSA lock and the ammunition in a factory box. Check with your airline for their packing regulations as they all do it slightly differently with the ammo transport piece.
One more warning. Look up the statutes for handgun possession in DC. As far as I can read, it is illegal but I am not an attorney. I'd hate to not hear back from you for a few years.