A modern natural history museum located in a freight elevator dedicated to its signature curatorial style of "Object Journalism" that draws parallels to the older cabinet of curiosities model.
The Brooklyn Flea is another treasure trove of odd finds, cool artifacts, and handmade goodness. If you're looking for inspiration or just something cool to hang on your wall, you can probably find something here. And if all that treasure hunting leaves you hungry, you are in luck--there's a delicious assortment of local food vendors and artisans there too.
The best underground shows you will find will have free pbr and models as far as the eye can see. They are rarely announced, so keep your eyes out for promising openings, especially with Milk Underground coming up.
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.