GREAT Idea 
Of course, I welcome you to the big New York City, which everyone deserves to visit a dozen times in their 20's and then sparingly in their 30s and on.
I'm originally from NYC, born out in Jamaica Queens. I think everyone knows about Manhattan, which we call "the city" if we're in other parts of New York City.
That's right--there are five main "hoods," each with dozens of smaller neighborhoods. We call them the five boroughs:
Manhattan, Brooklyn (the fist two to visit), Queens, The Bronx and Staten Island, as ordered on the map above.
This is also the unofficial importance of each part of NYC and for the purposes of your trip, you can stick to the green and yellow regions.
It's not possible for me to outshine the hundreds of travel help on the internet. There's a lot of information out there but I can definitely help you go deeper in hobbies or pursuits you enjoy the most.
Plus, I can offer the following real talk tips:
Rich people live in New York City, usually just a few streets away from people who are not very rich at all. So when it comes to buying things, you always have a choice between where people with lots of money shop and where people without lots of money shop. Teachers don't make a lot of money for the most part and are nice people to talk to.
Wear comfortable shoes. We walk a lot here, especially when traffic lights and one way streets will double the time it takes to walk five blocks. If it's further than that, consider the subway or a taxi, depending on which is easier.
You always have to pump your own gas. Ain't no full service. Plus that little stand that keeps the pump on hands free doesn't exist in the five boroughs.
NYC folk enjoy tourists and will talk to you even if their face has a (I smell something) stank look.
Enjoy your travel plans!
RE: Steemians! I need your help!