Quantico is just off I-95, but it also has its own rail station on Amtrak (courtesy of military family needs). The closest and most convenient airport is Ronald Reagan National Airport IATA: DCA. The other option is to take the Metro to the Franconia-Springfield station  on the Blue Line, but Amtrak is more convenient.
For the Marine Corps Museum, you would best take a taxi from either the Amtrak station or the (farther away) Franconia-Sprinfield station. Bus service is available via OmniRide , but service is limited, and stranding is likely.