I've only been for work so my excursions are pretty limited, but I agree with James there is a ton to do.
If you're not climbing the statue of liberty you can take the station island fairy for free, it goes right by the statue. Just ride over, and come back. The 9/11 memorial museum is a must. Plan for a couple hours, more if you want to see everything. And bring tissues, not ashamed to say I was moved. The observation deck is cool butnothing mind blowing. I was only able to go at night near 11 which meant no line and cool views of the lights.
You can usually grab last minute tickets to shows, but not always. Even discounted they'll be pricey.
Don't be afraid of the subway. It's my preferred transport and something you should experience. Uber is a good alternative.
I have a few favorites for pizzas and burgers, but there are a lot of fancy places to eat too.
None of the airports are great but Newark is the one I prefer. You can stay in jersey and take the ferry or path to the city. But staying in Manhattan is a different experience.
Also, it smells in the city. Garbage gets stacked on the curb and somethings always under construction. It's not as glamorous as the movies but cool to experience how different a culture can be from how we live.