"Mills" is my local bar, I live in that block, it's an old horse stable (for two horses) it's tiny, cozy, great drinks, food is great too - in the summer you can sit outside
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!
When I first visited NY in 2010 the New Museum left a permanent impression. I love the architecture and the concept of the piled-white-shoe-box look-a-like building at the edge of the Lower East Side neighborhood. The exhibition program is appealingly different, and the bookstore in the foyer is not to be missed!
The best place to see a dramatic sunset is right by Valentino Pier in my neighborhood of Red Hook, Brooklyn. This was right after a storm. It's wild and brilliant and peaceful all at once.