The Museum of Natural History!!!
American Museum of Natural History
Mango is a great one to tell you about NYC. She knows it really well.
I sent Rita an email already.
FYI--Many of the museums, including MOMA, have free entrance on Friday evenings. The Staten Island Ferry is free also. It's a great way to see the Statue of Liberty and Ellis Island. if you're doing the 5th Ave tour, Trump Tower is free to go in and one of the nicest bathroom stops, if you dig marble and lots of it.

Across from FAO Schwartz is a small park like sitting area with fountains and in front of FAO. Bring your lunch and enjoy. I like the one across the street because you can watch the coming and goings of Plaza guests, the limos, Bentley, etc. You can also go in the Plaza- their entrance is spectacular, especially around the Holidays.
Relatively inexpensive, Top of the Rock. best views and you can combine with an NBC tour. Quite often, you also get to sit in on a recorded show for free in addition to the tour- total for both- about $30.
Broadway on a Budget-- Haven't done this personally, but know people who have, do stand by. if they have extra seats available, you can see a show at a reduced price.
Check for free concerts in any of the parks. Another fun thing to do.
The Museum of Natural History-- during holidays and spring breaks, it is a mob scene! If you hate crowds, do not go there. Also, plan on spending an entire day there if you do go and wear good walking shoes.
But, by far, my favorite place to go is Wall Street. The architecture and history behind the buildings is quite interesting and spectacular. There's also a feeling there you can't match anywhere else in the world. This can be done after the free ferry ride. South Street Seaport is nearby and there's always something fun going on there towards the evening and lots of choices for seafood that just came in fresh. Bodies, the exhibition is around that area. You have to pay, but I hear it's great.
if you like people watching, Greenwich Village and cultural watching, Chinatown and Little Italy. Quite often, on the weekends, there are lots of people in the street and lots of ya ta ya ta going on and eating cooking on the streets in Little Italy. Check before you go. They are like mini San Gennaro Feasts. One could eat their way through Little Italy inexpensively.