I lived in DC most of my life and did not see everything. The best plan is to pick what you really want to see, what you can't see anywhere else, and do that. I would rank monuments above museums, and only pick two or three museums to tour. After the first couple, staring at things can get rather boring.
Highlights: Botanical Gardens; National Zoo; Air and Space Museum; Washington, Jefferson, Lincoln, and Roosevelt Memorials. I like the National and Hirshhorn Galleries but I'm an art junky.
If you get a chance, visit the Iwo Jima memorial at sunset. Even better if there is a full moon.
If you like beautiful buildings, visit the old executive office building, the old post office, union station, and the Library of Congress.
If you want to see important paper (Declaration, Constitution, etc.) visit the Archives.
Just what I have listed above will easily fill 4 days. And I haven't mentioned the majority of museums...
Definitely use metro. Cherry Hill is the most stress free way to visit in an RV.