We take our church youth group each year to a retreat program in NYC that we follow up with what we call the "Manhattan in a day" tour. Alot in a day (we start 10am and get done around 7pm)- but it's all pretty much free/low cost. (One of the youth group dads is from NYC and leads us all around.)
We take the Subway from the church to Times Square. Let them go through stores, people watch, play with the Forever 21 billboard, lol. We grab something quick for lunch in this area mainly because it ends up being lunchtime.
Walk to Rockefeller Center. Let the kids walk around whatever stores interest them, see the plaza, etc. (We don't, but you could go up Top of the Rock. It's a significantly faster wait than Empire State Building. {A friend and I went Memorial Day weekend- right up TotR, 2 hour wait for ESB.})
Walk up 5th Ave- see more stores, look at Empire State Building and go into the NY Public Library. (The reading room upstairs is neat.)
Hop on subway to former World Trade Center site. Walk around- when we went in April, there was a temporary museum. I think the real one is open now. (As is, of course, the memorial.) Check out Trinity Church/ Wall St.
Walk to Staten Island Ferry- take over and back to see the skyline and Statue of Liberty.
I'm sure there is a subway or something available, but we then walk to Chinatown (through the courts- because our guide is an attorney and wants to show where he used to work.) Grab some supper there and then subway back to Upper East Side (where the church is/we continue the night program there and spend the night.)