The first place to start when it comes to museums in Washington DC is the National Mall. Europeans will tend to think of the Mall as a gigantic shopping complex, but this is actually a strip of grass between the Capitol Hill and the George Washington Monument (the distances in the US tend to be larger that Europeans might be used to, so ‘a strip’ is really a pretty big chunk of land). The museums here are part of the Smithsonian Institute and, as such, are free of charge. A couple of the most interesting ones are:
The National Gallery of Art. The NGA can rival any art collection in the world and it was formed mostly from individual donations. I would recommend the Monet (and, in general, the impressionist collection, including Degas) and Rembrandt collections, which are extensive.
The Air and Space Museum. The cockpit of a Boeing 747 that you can visit is one of the attractions, but there is also an entire room dedicated to the Wright Brothers and their first flight. As in many American museums, you get a chance to experience first hand some of the levers and materials used for the first flight.
The National History Museum. This museum has undergone an extensive renovation and has opened to the public less than a year ago. If you have seen the movie “Julie and Julia”, then this is the place to visit a reconstruction of Julia Child’s kitchen, made to scale.
National Museum of Natural History. This is a comprehensive museum and everything from dinosaurs to the oceans exhibit is worth a site. Notably, you should go see the mineral section, with some of the largest precious stones in the world, including the Hope Diamond, deep blue in color and, allegedly, also cursed.
National Museum of the American Indian. This museum is not that impressive in terms of exhibited items, but there is extensive information, so the visitor can learn about a variety of characteristics of American Indian people throughout the Western Hemisphere, from religion to traditions and mythology.
Other museums on the National Mall: Arts and Industries, Hirshhorn Museum (lots of modern sculpture, so, only if you are really in the mood for something like this) and Museum of African Art (a small, but worthwhile museum).