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 definitely Ronald Reagan National Airport IATA: DCA. The other option is to take the Metro down 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.
National Marine Corps Museum, 18900 US-1, ☎ +1 800 397-7585 (email@example.com), . 9AM-5PM daily. Believe it or not, this is one of the best exhibited museums in the area, including D.C. proper. It is dedicated to Marine Corps history, but also American history more generally. Some of the exhibits are not very child-friendly, with more graphic and violent imagery.
Tun Tavern, 18900 US-1, ☎ +1 703 649-2369, . The most fun restaurant in the area for visitors is actually in the Marine Corps Museum! (The mess hall can be a cool option, too, though.) The food is straight-forward American, be it soups, salads, entrees, etc.$10-25, children's menu available.
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