You don't need a car - the subway system is very easy to use and gets you just about everywhere you'd want to go.
There is so much to see and do in Washington, and a lot of it is free. You can tour the Capitol building (free), visit all of the Smithsonian museums (free), visit the National Zoo (free), visit the monuments (free). There are certainly paid tours that you can also take - there are bus tours of the monuments, for example. It's worth it to visit Arlington Cemetery and the Georegetown area is also neat to walk around and explore (lots of good local places to eat there too). There are plenty of others museums and sights as well - Newseum, Holocaust Museum, Spy Museum, White House, etc.