Living in New York for over a year now, I have had the pleasure to help friends organize their itinerary. People often ask me whether they should watch The Lion King or Aladdin, which skyscraper observatory has the best sight or tips to find the cheapest places to buy clothes or electronics. Sadly, almost all of them fail in one aspect: they never include the other boroughs on touring list. 
New York consists of five boroughs: Manhattan, Brooklyn, Queens, The Bronx and Staten Island. Even though Manhattan is obviously the most popular due to its cultural, commercial and financial importance, the other areas have a lot to offer. Make an effort and try to add some of the spots I list below. By doing so, you will a get better picture of what New York really looks like. 
The most populous and diverse borough has so much to do and see, you could dedicate just one trip to Brooklyn. Some visitors think just going on a stroll along Brooklyn Bridge is enough, however they don’t know how much they are missing out. Graffiti in Bushwick, cafés and bars in Williamsburg, Coney Island’s amusement park and beach, Prospect Park, Botanical Garden, Christmas decorated houses in Dyker Heights and so on. Not taking some time to go sightseeing here is a big mistake. 
Geographically speaking this is the largest NYC borough. First of all, go to the Gantry Plaza State Park to see an amazing view of Manhattan. Then, check out the Unisphere, a giant globe made of steel located at Corona Park. Attend some interactive exhibitions at the Museum of the Moving Image or learn more about art at MoMA PS1.
The Bronx
This is not as dangerous as most outsiders think. Things have changed as being a resident in the city costs a lot. Visit the Bronx Zoo, Van Cortlandt Park or Pelham Bay Park, Botanical Garden or watch a baseball match at the Yankee stadium.
Staten Island
Probably the underdog of all five boroughs, Staten Island should be on your list as well. Ride a bike along the Verazzano Narrows Bridge, take a ferry to look at the Statue of Liberty for free, go to Staten Island zoo or pay your respects at the September 11 Memorial.