I live above TriBeCa and take walks into and around this area often. It's quiet, usually on the emptier side, and hosts a few gems of restaurants including Tiny's, Takahachi, and Odeon - all on the same block of West Broadway.
The New York Earth Room is an interior sculpture by the artist Walter de Maria that has been installed in a loft at 141 Wooster Street since 1977. Go see!
Hands down the best wood fire pizza I've had in Brooklyn. Great for a date or if you're feeling fancy schmancy. (Also, don't sleep on the caesar salad here.)