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Washington District of Columbia is the capital city of the United States, located on the East coast. It forms its own federal territory and is not part of the surrounding states.
South towards Richmond
Take a yellow subway line toward Huntington station and get off at Pentagon station. From there take a bus 21A, 7A, 7F as closer as u can to the King street, and then walk along toward 395 highway, you will pass a gas station and you can start to hitchhike before you will see a sign saying Richmond. That's where you need to go. It's enough place for a car to stop and the traffic is not too crowded.
East towards Annapolis, Delmarva Peninsula
Just catch the orange line to new carollton as it will get you right next to a fairly busy ramp onto the interstate 50. You can also catch one of the buses that leaves from there and goes to Bowie, but there is much less traffic
West towards Knoxville
This is good for getting on to Interstate 81, a popular truck route. Take the orange line all the way out to Vienna, the last stop westbound. From here, catch the Fairfax Connector bus to Centreville - all the buses numbered between 632 and 644 go there, but stop in different places close to the highway. Check with the bus driver in advance to make sure where he can drop you off. The on-ramp from Sully Rd onto I-66 is the connector for Dulles Airport, so it is popular with folks going a fair distance, but there's no official bus stop next to it. So sometimes the driver will let you off right near the highway, other times you'll need to walk. If you prefer not to walk, or not to wait and pay for the bus (the buses aren't frequent at all, maybe once an hour), you can try to hitch from outside Vienna metro station itself, but it's not the best spot. There are a pair of truck stops once you hit Interstate 81, just south of the town of Strasburg
North towards Baltimore (Maryland) / New York City
One option for I-95 is: Take the metro to Forest Glen and try to catch someone getting on 495 East on their way to 95. There's no room for someone to pull over on the north side of 495 where the metro station is, but there's a Lutheran church on the south side with a parking lot right before the on-ramp (you'll need to go past it a ways to give drivers time to stop)