In no particular order:
-- See who's playing in town that week (bands)
-- See what's playing at the Angelika
-- Go to at least one of the following: the Met, the MoMA, the Guggenheim
-- Spend some time loafing in Central Park and on the High Line, just 'cause they're lovely
-- If you have ANY interest at all in photography, pack extra batteries and memory cards. Between people (pretty much everywhere) and architecture (besides the iconic structures like the NYPL, Empire State Building, Rockefeller Center, et. al., you could spend your whole week just on architectural photography)
-- If it's still there, the Tibetan restaurant on Houston near the Bowery Ballroom is quite good
-- Do not, under any circumstances, let anyone convince you to get a sandwich at the Carnegie Deli. The pastrami at Katz's is better.
-- If, on the other hand, you're not into meat, there are some great vegetarian and vegan places around Union Square.
-- Corollary 1 to the above point: If walking through Union Square Park, you will be given approximately one ream of flyers, and asked to sign roughly 57 petitions.
-- Corollary 2 to the above point: If you are in the area of Union Square, you really should go to Forbidden Planet and The Strand, the former for its graphic novel and geek paraphernalia selection, and the latter for a selection of books that more than makes up for the mostly ineffectual staff.
-- No visit to New York is complete without buying food from a dodgy-looking grease truck. If you're a vegetarian, don't despair; there are also dodgy-looking vegetarian options as well (and I'm sure there's some kind of app for finding them)
-- If you're looking for antiques in New York, the best place to look is New Jersey or the Hudson Valley region of NY. As with everything else in NYC, antiques sell far above what they should.
Will add more as I think of it.