Just to name a few of my favorites:

• Greenwich Village
  
• Chinatown New York
  
• The Theater District - Times Square
  
• The Central Park Boathouse at Central Park
  
• Fifth Avenue
  
• The Flatiron Building
  
• The Metropolitan Museum of Art
  
• Museum of Modern Art
  
• High Line Park
  

Early one morning, grab a coffee (perhaps from One Girl Cookies) and take a walk around Dumbo in  Brooklyn. I can never get over the view of Manhattan from  Brooklyn Bridge Park. In Manhattan, I think  Top of the Rock is a nice alternative to the Empire State Building because you get a really lovely view of Central Park (and you can get the ESB in your photos)! The other suggestions from Soleil and Drew are spot on.

Breakfast at  Balthazar Restaurant is a classic.  You'll see the New York digerati tuck in to pain au chocolat.

High Line Park.  Take a walk and see what all the fuss is about.

If you want to see the  Statue of Liberty, I'd definitely recommend taking the  Staten Island Ferry. It's free and a great way to see her.

Cross  Brooklyn Bridge, stroll around  Central Park, walk along the  High Line Park.

I also thought the  9/11 Memorial Visitor Center was quite impressive but I did find lines quite long except for quite early in the morning.

I'd say visits to  Guggenheim Museum,  MOMA and  Empire State Building are musts for a first visit to NYC.