I will always recommend the Natural History Museum -- I contributed to the Hyperbolic Crochet Community Reef
that is in the newly redone Sant Ocean Hall. There are a few surprises, like the bull mummy in the Ancient History wing, and the Stellar Sea Cow skeleton in the Osteology hall that I think are well worth looking for.
You should eat at least one time at Mitsitam Cafe
in the National Museum of the American Indian. The food is amazing -- it consistently gets great reviews from the foodies.
And don't miss out on Millenium Stage
-- there is a free performance every day at 6pm at the Kennedy Center. Go to Foggy Bottom Metro and take the free red shuttle there and back.
I'm rather fond of the National Postal Museum,
too. It's right beside Union Station, and since it is part of the Smithsonian, absolutely free.
And since you're going to be in Dupont Circle, I recommend eating at Teaism
at least once.