How long are you going for?
We have just got back. We bought the 5 day New York Pass www.newyorkpass.com/new-york-pass-prices.php
We went to so much - when I got back, I sat down and worked it out - if we paid for everything we did separately it would have cost us more than $600 each (and we paid $250 each for the pass) - so a massive saving. And we didn't have to worry about not being able to do stuff when we were there.
You can save money on food - there are 99 cents pizza places EVERYWHERE - 1 or 2 slices of pizza is more than enough for tea. We only ate breakfast and tea. A couple of days we maybe stopped for a coffee and cake if we were peckish.
As others have mentioned, Staten Island Ferry is free - we went to Staten Island Zoo which is included in the pass, as a man outside Central Park Zoo told us it wasn't worth the money.
Walking Brooklyn Bridge is free - DS loved being in Brooklyn because of Brooklyn 99 haha!
We did the Liberty Cruise tour (Circle Line Ferries - included on the pass), but you can see the Statue of Liberty better on the Staten Island Ferry.
Walk the High-Line, it's an old railroad which they are turning into a park. That is free.
9/11 memorial museum was good - I enjoyed that more than I thought I would.
There is an interactive tour bus called 'The Ride' (also included in the pass), which is well worth doing - very entertaining, and my teenager would love it. I think it's $69, so quite steep for 1.5 hours if you don't get the pass. experiencetheride.com/
Other highlights for us including Empire State Building (we went first thing in the morning to avoid the queues) and Top of the Rock (which we did at night). 9/11 memorial museum was touching in a way I didn't expect, and I'm really pleased we went. Loved the Intrepid Museum. Also this place natgeoencounter.com/
And we went to a catacomb candlelight tour, which particularly impressed teenage DS - the tales that the guide told were great, all about the 'Gangs of New York' etc ...
Sorry, I realise I've just typed an essay :(