I'm a big fan of taking a walk or boat trip along the C&O Canal or the Potomac River. Two good places to rent canoes or kayaks are Jacks and Fletcher's Boat House. If you will have a car with you on your trip, a great place to go is Great Falls Park. There you can walk the canal or take a really short hike to a falls overlook.
For local flair, I like Eastern Market. There is a market building with food vendors that is open all week. However, on weekends, there is a farmers' market, craft fair, and flea market vibe that comes to the sidewalks surrounding the market building. If you're in the area, it's also worthwhile to browse the shops on 8th Street SE between Pennsylvania Avenue and I Street (Barracks Row) and grab a bite to eat at one of the myriad fabulous ethnic restaurants on that stretch.
A uniquely DC treat is to attend an event at a foreign embassy. Check out the
Maison Francaise website to see if they are offering any concerts or events while you're in town. If you're willing to give up your email address, you can join the
International Club of DC and attend one of their monthly embassy receptions.
You might be interested in the book
Off the Beaten Path: Washington, DC. Though not a guide to living like a local, it will direct you to interesting attractions other than the monuments, Smithsonian, U.S. Capitol, and White House.