I've lived in DC for the last 12 years. There are a ton of good ideas on this thread, and I think cover most of what you might want to see. My advice: bring comfortable walking shoes - the city is very walkable. If you are visiting in summer, bring water bottles too - the heat and humidity can be brutal if you are outside all day. Metro is great (when it's working properly) and there are tons of Ubers here too if you need one.
Some of my favorites places I take friends and family that visit are:
1) National Mall - walk the monuments (Washington, Lincoln, FDR, Jefferson, WWII, WWI, Vietnam, Korea, MLK, Reflecting Pool (lots of walking, but it's great)
2) Arlington National Cemetery - IMO the most sacred place in DC. So humbling.
3) Smithsonian Museums (American History, Air and Space are my favorites) - do the Air and Space at Dulles too, that's where they keep the big planes and a shuttle
4) Tour of Capitol building - get in touch with your congressman or senator for that
5) Ford's Theater - where Lincoln was assassinated. They do plays and performances still there - look into tickets. They also do tours and you can see the box where Lincoln was shot.
6) Mount Vernon - George Washington's home. On the potomac - absolutely beautiful. Has a distillery there as well.
7) Gettysburg - about a 90 minute drive, but worth every minute. So much history there. Plan a full day for this.
8) Trump Hotel (Formerly the Old Post Office) - Like Trump or not, it is a beautiful building. You can get a drink in the lobby bar and just look at the architecture. If you do like Trump, it's a must.
9) Library of Congress - probably the most beautiful building in DC and very underrated. Take a tour.
10) Kennedy Center - they do ticketed shows and also have free performances every day. Check out the schedule. Also, go up to the rooftop for a GREAT view of the city and Potomac.
11) Georgetown waterfront - great restaurants right on the river, can sit outside and have a drink.
12) Nationals game - if you're into baseball, it's fun. The Nats suck this year, so tickets should be cheap.
I'd also recommend going to eat at Old Ebbitt Grill. It's been around DC since the 1880's and is a staple. Get a reservation in advance. Walking in will likely result in an hour wait during lunch and dinner times.
Other good places to eat that are pretty reasonable are: The Hamilton, Ollies Trolley, District Taco, Off The Record (Bar at The Hay Adams Hotel), Hill Country BBQ.
If you want any other ideas of good restaurants, or hidden gems, let me know! Happy to help out fellow Clones coming to visit DC.