Actually, New York City has more neighborhoods than you need to know. But if you see this list more or less If you learn by heart, you will come a long way.

Neighborhoods Manhattan: 

Harlem Covering virtually all of Upper Manhattan, it is the island's largest neighborhood above Central Park. The center of Harlem is to the west.
Uptown Around Central Park above 59th street. On the left the Upper West Side, on the right the Upper East Side where the richest New Yorkers traditionally live.
Midtown Below 59th Street and above 14th Street. Here you will find Times Square, Bryant Park, the major retail chains and food trucks in front of the offices around 52th Street.
Chelsea Part of Midtown against the Hudson River to the west. The indoor hall, the Chelsea Market, has been a highlight for foodies for years.
Union Square. Midtown east of Chelsea.
Downtown Everything below 14th Street.
Greenwich Village aka The Village. Officially the entire area between 14th Street and Houston Street, but actually Greenwich is mainly referred to West Village indicated. The neighborhood east of Broadway is obviously the East Village.
Meatpacking District Small harbor area on the Hudson in Greenwich, right below Chelsea. Once the hippest part of the city and still not to be missed, although nowadays the area is mainly populated by tourists and it is expensive.
SoHo either South of Houston (River). Thanks to the small streets and boutiques, this is the most famous shopping area in Manhattan, right below the West Village.
TriBeCa or Triangle Below Canal (Street) and the neighborhood immediately below SoHo. A residential area with a top restaurant here and there.
Lower East Side (LES) is the neighborhood where hip Manhattan now hangs out for cocktails and late night dinners.
Little Italy en Chinatown Busy and noisy but oh so cozy. Below each other, Little Italy above, China Town below, SoHo and TriBeCa on the right, LES on the left.
Nolita either North or Little Italy.

Neighborhoods Brooklyn: 

Williamsburg Brooklyn is three times the size of Manhattan and yet many tourists think they have seen it all in North Williamsburg. The hip neighborhood is located directly under the Williamsburg Bridge and is home to Beyonce, Jay-Z and Blue Ivy. And from many Orthodox Jews, especially in South Williamsburg. Take a bike from the Lower East Side.
Greenpoint North of Williamsburg, the two neighborhoods flow effortlessly into each other. Northeastern Greenpoint borders Queens.
Bushwick Raw, creative neighborhood in the southeast bordering Williamsburg.
Dumbo Right under the Brooklyn Bridge. The neighborhood borders the Hudson and is always full of tourists taking photos of the Manhattan skyline.
Brooklyn Heights Residential area with the well-known brownstones, the typical New York houses with steps at the door.
downtown brooklyn East of Dumbo and Brooklyn Heights and the official center of Brooklyn.
Cobble Hill, Boerum Hill, Prospect Park en Park slope Adjacent neighborhoods south of Downtown Brooklyn where more and more hip cafes and restaurants are slowly opening their doors.

And if you think you need more accurate help on the spot, add this list to your favorites:

Op urbanoyster.com you will find a good selection of things to do and where in the city. The New Yorkers behind it organize original tours through the city.

Download a city map via the app City Maps 2Go, also works offline.

Through which channels you reach those people, classic and out of the box. urbanspacenyc.com you will find pop-up food markets appearing here and there, often combined with festivals, such as Smorgasburg, Brooklyn's food market.

You're welcome.