Here is a list of neighborhoods to explore

 

Chelsea = art galleries - see info about openings in NewYorker.com

Tribeca

West Village

East Village

NYU (between the E and W villages)

Flatiron

Soho

Rockefeller Center

Lincoln Center

Madison Ave above 59 Street (fancy shops)

Museum Mile (5th Ave adjacent to park)

Upper W Side

Upper E Side $$$

Central Park

DUMBO (Brooklyn)

Williamsburg (Brooklyn)

Bushwick (Brooklyn)

Brooklyn Heights and the promenade

Park Slope (green and pretty; brownstones)

Red Hook (Brooklyn)

Chinatown in Flushing Queens

Jackson Ave Queens

Hudson Yards (north end of highline)

Coney Island

Times Square at night