Some of the museums you want to go to besides the MET are The Museum of Modern Art (MoMa), the American Indian Museum, the American Museum of Natural History and the Guggenheim Museum.
The MoMa (http://www.moma.org/) has everything from paintings to sculptures and drawings to photographs, prints and architectural models.
The American Indian Museum (http://www.nmai.si.edu/) has exhibitions, dances, music and films that teach and portray the Native American history and heritage.
The American Museum of Natural History (http://www.amnh.org/) has collections and exhibitions spread out on four floors that illustrate the creation of everything on earth up to the present day. Artifacts represent different cultures, sciences and historical times.
The last museum we mentioned is the Guggenheim (http://www.guggenheim.org/) which is where you will find art work from artists of the 19th and 20th century.