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.
This is a pro tip, but also a summer tip, take the ferry from Lower Manhattan to Rockaway and spent the day. The beach is great and the area is impossible to gasp that it's 30 min from Manhattan (without traffic) - Eat as many tacos you can, cause it's the best in the state of New York!
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.