The permanent piece called "Meeting" at MoMA PS1 is an etherial experience of light and color and quiet. Sit and look through the cutout in the ceiling and watch the light change both outside and in. You can't tell what you're looking at until maybe a bird flies by.
I can’t get enough of the dioramas, but my favorite parts of the AMNH are the Halls of Meteorites and Minerals. There are some beautiful specimens here, and best of all you can actually lay your hands on the 4.5-billion-year-old Cape York Meteorite.
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.