Air & Space is supergreat if you like air or space. But, obviously, it’s narrowly focused on those pursuits.
Natural History has a newish oceans exhibit that’s quite rad, and I love the mammals exhibit on the other side of the first floor. I haven’t been to the new Human Origins exhibit there. It’s supposed to be good, although there is controversy as it is funded by David Koch, one of the infamous Koch brothers who fund the Tea Party, climate-change denial organizations, and right-wing think tanks; there has been some commentary that the exhibit itself downplays the significance of climate change. The rest of the museum is IMO inconsistent, but there is a pretty cool bug zoo upstairs.
American History has a lot of eclectic cool stuff as well, but I don’t think I’d send someone there off the bat. I just wanted to bring it up so I could casually mention that I once had a photograph on display there.
For fine art the National Gallery is a good bet, especially if there’s an interesting show on. But if you’re looking to do a small museum for an hour or so after spending most of the day elsewhere on the Mall, either the Sackler (rotating exhibits of Asian art) or esp. the Freer are excellent choices. The Freer is interesting – it’s got Asian art intermixed with paintings by James MacNeil Whistler and his circle. They were very interested in Asian art, and so they got their patron (Charles Freer) to begin collecting it.
Also excellent is the Portrait Gallery/American Art Museum (two museums, one building). It’s a bit off the Mall but easily accessible by Metro across the street from the Gallery Place stop.
To boil it down: Air & Space unless that leaves you completely cold. Definitely stuff to see at Natural History. Then I’d send you to the Freer, and/or the Portrait Gallery, but YMMV.
Finally, it’s worthwhile to check the Smithsonian website for the particular exhibits that will be up when you’re in town. There might be something so cool that it gets priority – for instance, if you’re coming this weekend, you’ll definitely want to hit the Sackler to see Hokusai’s 36 Views of Mt Fuji (and thanks for reminding me – I’ve got to get my ass down there this week), but it closes Sunday. Due to the sheer size of the Institution, the website is by necessity pretty damn overwhelming, but it’s still a useful tool.
–Cliffy
P.S. Skip the Spy Museum. It’s not awful, but it’s not great, and it costs $20. Compared to being able to see the Apollo 11 capsule or Gilbert Stuart’s Lansdowne portrait of George Washington for free? Hell naw.