Depending on the airport, check with security and airport management, and show them your costume if need be, to let them know when you will be in suit and where in the airport.
If you're going to fly on plane in suit, then be prepared to be *IN* suit, all through the flight, to the next airport, if you're allowed.
However, since the terrorist attack of September 11, 2001, you may never be allowed into the airport, or just be allowed into the non-security areas. Washington, DC's Reagan National Airport (DCA) does not even allow fursuits into the airport.