Federal law does not restrict those who can legally own a gun, from transporting legally acquired firearms across state lines for lawful purposes. No federal permit is required (or available) for the interstate transportation of firearms. But many states do, which of course differs from state to state.
While Federal Law on Transportation of Firearms applies in every United States jurisdiction, experience has shown that some jurisdictions provide particular challenges to those transporting firearms. This is why it is important to know the local laws and which particular states may apply.
Here is another helpful map tool from the NRA and is a guide to Interstate Right To Carry Reciprocity and Recognition
The laws are always changing. Do not rely on the map image above, check your state for current laws
Here are some specific links that applies to airplane travel
TSA (Transportation Security Administration) http://www.tsa.gov/