There are a tons of brilliant things to do in the big city; from hip art galleries to amazing restaurants. The most international city in the world that always offers pop-up-infested neighborhoods such as Soho and West Village, will make you feel like at home!
During my four years of College in Philadelphia, I used to visit New York City twice a month. The day always started at Sant Ambroeus, a coffe place that has become as iconic in New York as it is in Milan! ... we all know how Americans drink their coffee, but this special pink coffee place will make you feel like you have never left home. Your cappuccino will be as authentic as the one you drink in Corso Matteotti, Milano.
Then, you should head to the High Line, a "public park built on a historic freight rail line elevated above the streets on Manhattan's West Side". Here is where you can walk through gardens, view art and connect with neighbors while enjoying a unique perspective of the city.
After a visit around Hudson Yard, including the Vessel and the indoor shopping, you can head to a museum. The amount of museums in the city can blow your mind.... Moma, the Met, the New Museum, the American Museum of Natural History, the Guggenheim and The Guggenheim are just few of the ones you should check on.
Lunch at La Mercerie, a beautiful all-day cafe housed in a furniture store by Roman and William located in Soho.
Get lost between the boutiques in West Village where you can act as Carrie Bradshaw. Then go get a cocktail to a famous rooftop with stellar skyline views; you should check out Mr. Purple, Jimmy and Le Bain... the atmosphere is quite young and swanky.
If you are open to international dishes, definitely head to Jeju Noodle Bar, a Korean restaurant located in West Village that won a Micheline star for offering delicious ramen and non-noodle dishes that will make you go crazy. This place is affordable but does not really offer reservation... so you should just walk in and wait... but it will be deserved!