Going to museums like that is honestly one of the main reasons I'd like to visit the US. Unfortunately though, even just the biggest ones seem to be way too spread out to attempt in one trip, coming from Europe.
Fly into New York City. Visit museums there for several days. See a Broadway show. See the Intrepid.
Take the train down to Washington DC. Several days at the Smithsonian there, including a trip out to the Udvar-Hazy Center. Take a day trip by train to Baltimore to see the National Aquarium and the Maryland Science Center.
Rent a car, drive to a beach town in Virginia for a weekend; drive back and take the train back to NYC. Or, drive up to a Delaware beach instead and return the car in NYC.
See the other museums in NYC that you missed the first time. Fly home.
