My favorite museum in the Smithsonian - indeed, in all of Washington, DC - is definitely the National Museum of the American Indian
. I love that the outside of the beautiful natural landscape of the outside of the museum is part of the museum's exhibit, which demonstrates Native American tribes' connection to the natural landscape. I've seen baby ducklings swimming in the pond, heard a red-winged blackbird, watched dragonflies skate across a lily pond - and all within sight of the Capital building and the Washington Monument. The architecture of the Museum itself also suggests adobe buildings. There is also a garden along one side of the museum that uses the "three sisters" method to grow corn, squash, and beans, as well as tobacco and other crops. It's really cool. And that's just the outside.
I could go on and on about the exhibits, which make a point of (1) not showing artifacts plundered from the grave sites of Native American peoples; (2) showing Native American peoples as living and evolving peoples rather than somehow "trapped in time"; (3) showing beautiful depictions of Native American craftsmanship; (4) showing more than just the European angle. My favorite exhibit is probably the one on spirituality, which shows similarities and differences between the spiritual beliefs of a number of Native American tribes throughout North and South America. It's interactive, there's videos, it's very cool.
The cafeteria is also worth a visit, as there is a selection of delicious food that's representative of different regions of the Americas. You can get (depending on the season) tamales, salmon, bison burgers, squash and crab apple soup, various kinds of salads, yucca, yams...it's delish. I go here sometimes just for the outside of the museum and food. (It is a little pricey, but given what you're eating it's comparable to a nice restaurant. But definitely not as cheap as the McDonald's at the Air and Space Museum.)
My favorite part of the museum, though, is this statue by Apache artist Allan Houser. It's entitled Sacred Rain Arrow
, and it depicts a young man shooting an arrow into the sky as a prayer for rain for his people. To me, it is the essence of prayer. (This photo depicts the statue from before it was housed at the museum, because I was trying to find a public domain image.) Please note that sometimes the statue goes on tour, etc. or they stick it in a weird part of the museum so sometimes you have to ask the info desk about where he might be.