A direct quote from the NRA/IRA Guide to Interstate Transportation states:
Federal law does not restrict individuals (except convicted felons; persons under indictment for felonies; adjudicated “mental defectives” or those who have been involuntarily committed to mental institutions; illegal drug users; illegal aliens and most nonimmigrant aliens; dishonorably discharged veterans; those who have renounced their U.S. citizenship; fugitives from justice; persons convicted of misdemeanor crimes of domestic violence; and persons subject to domestic violence restraining orders) from transporting legally acquired firearms across state lines for lawful purposes. Therefore, no federal permit is required (or available) for the interstate transportation of firearms.
Each state has its own laws regarding the right to carry. Gary Slider of handgunlaw.us has an extremely informative and up to date website, with interactive maps and links to applications for permits in each state. This site is frequently updated and has a simple format. Gary answers every e-mail personally.
There is no uniform state transportation procedure for firearms. If in doubt, a traveler should carry firearms unloaded, locked in a case, and stored in an area (such as a trunk or attached toolbox) where they are inaccessible from a vehicle’s passenger compartment and not visible from outside the vehicle. Any ammunition should be stored in a separate locked container.
Authorities may search anywhere within your reach without a search warrant after a valid stop. They may not open and search closed luggage without probable cause to believe evidence of a crime will be found. This applies particularly to luggage in locked storage areas or the trunk of a vehicle. Consent is necessary for such search. You have the right not to consent.
You also have the right to remain silent.