Right on the New Brunswick Campus of Rutgers, the Zimmerli is particularly well known for its collection of Russian and Soviet avant-garde art, French nineteenth-century art and American nineteenth- and twentieth-century art.
This website lists the openings and ongoing shows going on that day in NYC. You can subscribe to the website to receive a weekly newsletter about art shows as well.
Google's tool for searching museums allows keyword searching (the results will be ranked using the Google ranking algorithm, not in alphabetical order) and index searching.
One of the world's greatest art collections, spanning from ancient to modern art. If you're interested in medieval art, make sure to check out the Cloisters.
MOMA has a great collection of modern and more contemporary art. Look at the bottom of the page for a link to PS1, an fantastic contemporary art exhibition space in Queens.
This museum of American art covers the 20th century to the present. The museum also hosts the Whitney Biennial, which showcases younger and lesser known artists.