In addition to the usual suspects (the Metropolitan Museum of Art, MOMA, the Whitney, the Frick, all magnificent) I want to point out a museum that is more off the beaten path: The Cloisters.
Take bus (the M4 **) or subway (A train to 190th St, then either walk in to the Cloisters through Fort Tryon Park or grab the M4 at the subway stop and ride) all the way to the northern tip of Manhattan Island, and view medieval art in a setting of natural beauty:
Located on four acres overlooking the Hudson River in northern Manhattan's Fort Tryon Park, the building incorporates elements from five medieval French cloisters--quadrangles enclosed by a roofed or vaulted passageway, or arcade--and from other monastic sites in southern France. Three of the cloisters reconstructed at the branch museum feature gardens planted according to horticultural information found in medieval treatises and poetry, garden documents and herbals, and medieval works of art, such as tapestries, stained-glass windows, and column capitals. Approximately five thousand works of art from medieval Europe, dating from about A.D. 800 with particular emphasis on the twelfth through fifteenth century, are exhibited in this unique and sympathetic context.Also... This one isn't an art museum, but it's a really really great museum that you can see relatively quickly: Lower East Side Tenement Museum.
Finally, the Louis Armstrong House Museum in Corona, Queens is a special treat on a Saturday, when senior docent Francis Lunzer is guiding tours. Francis, a retired jazz drummer and Armstrong enthusiast with an encyclopedic knowledge of music, does a tremendous job.
If you want to see a Broadway show while you're in town but aren't picky about which one, the TKTS booth (now with three Manhattan locations) is your friend: half-price same-day show tickets (the theater would rather get *some* money for the seats than none at all.)
More interesting and adventurous fare can be found off-Broadway (and off-off-, and off-off-off-, etc.) Buy a dead-tree copy of The New Yorker (inadequate online substitute for their events listings here) or Time Out New York a month before you leave and mark up the stuff you might want to see; buy another copy the week you're going to be there to validate that the shows you marked on the first list are still running. :-) (Obviously, this is also good practice for spotting any kind of cultural event in NYC, not just theater.)
Jazz clubs (major endangered species candidate) I have loved:
The Village Vanguard
Small's (late sets and 1:30 AM-til-whenever jam sessions on weekend nights)
** By the way, quick digression - the M4 bus is one of the best ways to see a *broad* swath of the City for cheap - just a $2 swipe of your Metrocard:
The M4 bus is one of Manhattan's best -- and most economical -- tourist activities. I enjoy riding the M4 sometimes just for fun because it offers an incredible cross-section of the variety of New York. It starts in bucolic Fort Tryon Park, runs through the Latin enclave of Washington Heights, skirts Harlem, traverses the intellectual (y)upper west side around Columbia, passes through the glamourous upper east side, and finally marches right thorugh the center of glorious midtown before terminating its run at Penn Station.