>Extra note, i'm staying in the Lower East Side but happy to
>travel around

Let's start from here. Hop on the F train, get off at York in Brooklyn, walk through Dumbo. Here you can get Grimaldi's...famous NYC pizza that people seem to love (me it's not worth it though). Right next door there is a Shake Shack...another popular nyc tourist destination until they opened up everywhere, that tends to have M-U-C-H shorter lines. This covers eating in the area, but the main reason why I recommend it is its got the best sightlines of NYC BY FUCKING FAR ...for example:

https://a2.muscache.com/locations/uploads/photo/image/383/0_4200_0_2800_one_manhattan-bridge-jbk018.jpg

Opinions on food in NYC's is a tough once since its the epitome of a foodie town. You're going to have people recommend you 3 michelin stars restaurants and hole in the wall ethnic joints. My favorite Soul Food joints are currently these two:

http://www.carlahallsouthernkitchen.com/

A little out of the way, but still close to Downtown Brooklyn.

http://streetbirdnyc.com/

One of Marcus Sammuelsson's restaurants in Harlem and my favorite...its a solid low in the cut time.

Speaking of Marcus's restaurants, might as well go to Red Rooster since its basically the big Harlem destination. While you're there, try to catch some jazz in the basement at Ginny's Supper Club. You can also go to the Studio Museum in Harlem to see art by black people (check to make sure if they've got exhibitions tho, last time I went they closed up most of the museum), and I would also recommend the Shomberg if they've got something happening that week:

https://www.nypl.org/about/locations/schomburg


BACK TO BROOKLYN:

Thanks to Dubai esque manhattan rents, it seems most of the big live venues outside of jazz spots have moved to BK. They are mainly roving parties trying to avoid getting too much attention but if you want to go listen to some solid techno, Output is great. And while there is a London outpost, Brooklyn Bowl is always good, particularly on a thursday.

GO TO QUEENS:
Flushing Queens is a cornucopia of great asian eats; its well worth a food trek out there.

TAKE A FERRY TO STATEN ISLAND
Just to go see the statue of liberty (not worth going actually inside) and it's free. Then hop right back in since that's all that's worth seeing there, lol.

After that, NYC's the kind of city where you want to stumble onto things. Id just figure out what neighborhoods you want to go to and just walk around anyplace that isn't Time Square.