See a Broadway show. You can do it for pretty cheap too with either the TKTS app or the booths.
Go to two Comedy Clubs. Comic Strip Live, and Comedy Cellar were my favorites. There are about 10 really good ones so just pick a few. Avoid the ones that offer you free tickets on the street though as most of them are hit and miss.
Obviously go to the WTC museum and the Statue of Liberty. Ride the Staten Island Ferry. It's free and a nice view.
Go to Times Square for an hour, realize it sucks, and then literally go anywhere else.
Go to the High Line if the weather is good. It's a former railroad that has been turned into a raised walking park.
The Financial district is a fun visit too if you love Capitalism.
On the Upper West Side, the subway will be your friend since you are a little far from most things. Uber or cabs work in a pinch.
Hit up museums too. Virtually all of them are good but that depends on your tastes.
The Intrepid Air Museum is good too.
You can get some good deals on night cruises with good views of the city and food. Groupon or Gilt City are good for that too.
Bars are literally everywhere. You don't really need to plan it. Just see a good one and go.
If you are fancy and want "clubs" head to the "Meatpacking district".
I would recommend doing a rooftop bar too. They are expensive but it is a cool experience.

For food, and I haven't been in a few years, is a place called "Hell's Kitchen" in Hell's Kitchen. It's not related to Gordon Ramsey.
Head to Little Italy and eat anywhere.
Eat Pizza, avoid the hot dogs.
Avoid Coney Island.

finally, we get to discuss diamonds

No Central Park?