As someone who has been to New York City several times, I have discovered a few things that I enjoyed so much that I did over and over again.
See a Broadway show
The Phantom of the Opera's most iconic scene
New York Times
No seriously, Broadway is unbelievable. I have seen quite a few Broadway shows, including Matilda, Newsies, and Phantom of the Opera. Matilda was the first one I saw and was amazing, but Phantom of the Opera blew me away. It left me in awe and even after years, I will never forget how breathtaking it was sitting there watching such moving and incredible performances by the most talented musicians, actors, singers, dancers, and the passionate work of everyone involved in the production. Broadway is well worth the price and your time. A trip to New York City is not complete without it.
Pro tip: if you want to meet the performers, immediately after the show, go find the stage door and ask them politely to sign your playbill or take a photo. Many performers are more than happy to do so- I did this with friends and a large crowd of scores of other people after Newsies. Just remember, they just gave it their all on a hot stage, so try not to take it personally if they're tired and leave right away.
Eat some Italian-American cuisine
A pie from John's
One of the best things about New York City is the food. While the Sabrett hotdogs sold on the street and at most tourist attractions are delicious and Junior's makes a mean cheesecake, nothing beats New York Italian food. In Manhattan, the best place for Italian cuisine is Tony's Di Napoli. The meals, which include pastas and salads are served on big platters which are meant to be shared between a few people, so if you're in a group or a family, it's the perfect place. Oh, and the cannolis are perfection. As for pizza, the best I've ever had in my life was from John's Pizza. John's has a few locations, but I go to the Times Square location which is situated in a former cathedral complete with a domed ceiling and elegant stained glass.
Pro tip: John's is super popular, so either come prepared to wait quite awhile (totally worth it) or call ahead if you have a very large group and make a reservation.
Go the New York Public Library
The Rose Main Reading Room
New York Public Library
Not only is the New York Public Library a gorgeous building with stunning architecture, it also is a great place to escape the elements (New York can get pretty hot during the summer and absolutely freezing during the winter). There are plenty of big comfy reading chairs, so sit down, take a breath, and take in the views from the windows, admire the exquisite artwork, or browse the collections. Both the interior and exterior look like something out of the Renaissance, so take the time to explore. You could literally spend several hours here. The authentic Winnie the Pooh stuffed animals have an exhibit in the children's section that is well worth the look for any Disneyphile.
Pro tip: Not as many places as you'd think in the city have public bathrooms, but the library does!
Visit the American Museum of Natural History
Visiting this museum was one of my favorite things I did in New York City. It was very quiet- a break from the rest of the city. Strolling through the museum was oddly relaxing. The exhibits were mesmerizing and informative. I literally spent hours here. This is a museum pretty much everyone would enjoy! A cool piece of trivia: the 2006 movie A Night at the Museum took place at this museum.
Pro tip: Buy your tickets online to save some time.
Head to a park and people watch
America is a melting pot, and New York is the epitome of this melting pot. New York City is an incredibly diverse city, filled with different cultures. Find a nice spot in a park and sit and watch all the people you see. At the very least, you'll see a variety of people from all around the world, and if you're lucky, you might spot a celebrity (I've spotted a few in New York), as many celebrities and famous people call this metropolis their home.
Pro tip: Bryant Park is a lot smaller than, but just as nice as Central Park.