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 Chelsea Market is filled with great restaurants and speciality food stores and pop up shops. I love stopping in and getting a gourmet grilled cheese from Lucy's Whey or a healthy juice from raw food restaurant One Lucky Duck, while shopping for dinner ingredients at the Italian market Buon Italia. For a great sit down restaurant, I love The Green Table, which serves great farm to table food.
The operators of the Peter Pan bakery make the best donuts I've ever tasted. This place is charming to its core. Decor and presentation: circa 1971 & virtually unchanged since.
Moma is worth a visit, but for a more lively Art experience ánd for free, one should visit Chelsea. A colony of 400 Art Galleries, showing the best in contemporary art. Running straight through is the high line, a 1 mile long linear park, build on a former railroad-spur, worth the walk.