Admittedly, there isn't much by way of epic rock climbing in the city of Manhattan. But if you have a love of bordering and puzzling out small problems, then you're in the right spot.
For the best/easiest-to-find bouldering, head to Central Park and look for the Cat and Rat rocks. Both are about 15 feet high, but the problems on them can be tricky enough that even an experienced climber won't feel let down. Plus, there are tons of fellow climbers around, so you'll be able to get advice and encouragement most times you're there. For a slightly less packed spot, check out the bouldering at High Bridge Park.
To log some gym time, Brooklyn Boulders is a great destination, with well-maintained and frequently changed problems on the many walls, funky colors and decor (think more graffiti than your average rock gym) and huge span of different people coming in to climb.
If you're a true climber, consider heading three hours north to The Gunks, one of the premier climbing destinations on the East Coast. You won't regret it.