Just thought I'd share our itiniary for our 3.5 days on a budget with children aged 10 and 13.
The first day we walked through the southern part of Central Park, then went to Battery Park and took the free ferry to Staten Island, taking in the views of The Statue of Liberty on the way. Staten Island itself was a funny little time capsule of a small town way out in the country so after having eaten our picnic we went back. Being in that area we looked at Wall Street though the kids found it boring.
The second day we went to the Apple Store by Central Park as that was a dear wish on behalf of the teenager. Next stop was The Brooklyn Bridge. Then another subway train and walking up Times Square - where I personally was going klaustrophobic from the crowds. But Bryant Park just before that was great. They even had an outdoor library. We went on to admire Grand Central Station before we called it a day and ordered Chinese take-out for dinner.
On the third day the kids were getting very tired from jetlag and walking. So we took it easy and browsed around Broadway near Penn Station, looking for souvenirs and having frequent stops. Then we walked The High Line Park and that was probably my best, at least most surprising experience - highly recommended. We ended up at Chelsea Market, also very nice.
On the last day we took the Roosevelt Island Tramway. That was fun. Not a whole lot to do on the island and the bus to the southern end had just left. But the ride and the views are fun. We took the subway to 110. St. to walk a bit in the nothern part of Central Park. We had hope for a vista view but didn't find it.
I loved how there were so many small spaces, many of them green, for sitting down and they were used by all. The subway was fast, cleaner than I thought and less crowded than the Tube in London. A great way to get around. Other than souvenirs we didn't do any shopping at all as we were going on to Portland, OR, with zero sales tax. It was very hot so I suggest bringing a hat in the summer. We found our accomodation through homeexchange.com where we have been members for many years.
Overall I was pleasantly surprised, loved what we saw and hope to return - with or without the family :-)