Monuments: I recommend renting a bicycle to see them all. The best time to do this is after dark, when heat and crowds lessen and the white stone is brightly lit. My favorite is Martin Luther King, Jr.'s, because visitors can read his powerful words and see his image striding towards justice.
Museums: Many of the Smithsonians are free! This is sadly rare here in the United States. I love it more for the way it values public good than I do for saving a few bucks. Some favorites are the Smithsonian Museum of Natural History (where I can nerd out about human prehistory as well as admire incredible gems) and the National Museum of the American Indian (emphasizes current-day tribes and nations, not just a static, historical over-generalization). The NAMI also has an amazing cafeteria with native foods from all over the Americas - best museum food, hands down.
Sense of belonging: Everyone is from somewhere, here! Moving to the DC area has been a balm for me after feeling like an outsider in Boston. As a world capital, we're connected to all points of the globe, with diplomats, immigrants, and intra-U.S. migrants alike. You're a "local" after roughly 5 weeks - you'll know it when you start staring at tourist groups instead of tourist sights. It does still blow my mind that this is also the home to so many institutions that were merely abstract, distant strings of letters before: USDA, EPA, FBI, CIA...
The Nats: Our baseball team is the Washington Nationals. The modern stadium is a great place to experience a true American tradition, complete with ballpark hot dogs, singing "Take Me Out to the Ball Game," and hilarious mascots. In our case, some of our most famous past presidents (well, their mascot counterparts) compete in a race around the bases between innings!
