OK, here are a few places to consider:
Bicycle Habitat--224 Lafayette St. (212) 431-3315
Small place in SoHo worth visiting just to meet Hal, resident Master Wrench and Enlightened Being.
http://bicyclehabitat.com/site/index.cfm
Conrad's Bike Shop--25 Tudor City Pl. (212) 697-6966
Although I don't really go to this place often since it's small and out of the way (in Tudor City a block away from the UN), I mention it because I saw Jerry Seinfeld order his Seven Odonata there a couple of years ago. Not much selection for parts and accessories actually, except for only the most high-end stuff imaginable, and as with most bike shops in NYC you'll pay at least full retail along with 8.625% sales tax.
R&A Cycles--105 5th Ave. Brooklyn (800) 825-3762
Pretty much The race shop in the area although it's not in Manhattan. Huge selection of everything.
http://www.racycles.com/
Of course there are a number of other shops around town but these are my personal favorites. If you're able to ride while you're here check out this site:
NYC Ride Library
I hope you enjoy your visit here. The weather hasn't been so great lately but we usually have more than a few beautiful days in April.