The metro is very good and not expensive. The Smithsonian is between the Capitol and Washington monument. Washington's monument is right across the street from the White house. Right next to the WM is the reflecting pool and the Vietnam memorial and Right next to that is the Lincoln memorial. All walking distance. It can take several days to see most of the Smithsonian. Mount Vernon is down the river and you may consider driving to that. There is little to no parking in the city so forget that. A great view is to sit on the top step center of the Lincoln memorial and look over the reflecting pool. And by the time you get there you will need a few minutes of rest.