I've only been to NYC once, but I thought that all of these were very worthwhile:
The Met - we could have spent an entire day here. It was really amazing to see all of these paintings and artifacts dating back thousands of years.
Museum of Natural History - We didn't have any kids with us, but this was also the kind of place that we could have spent all day.
Staten Island Ferry - This is free and gives you a great view of the Statue of Liberty. We only had 3 days in the city and I had heard that the lines to go to Liberty Island were very long, so we opted for the ferry instead.
Top of the Rock - Again, we heard that lines were long for the Empire State Building, so we opted to do the Rockefeller building instead. From up there, you can see the whole city, including great views of Central Park. From the observation deck, you can see the people over at the Empire State Building and I don't think their view could have been any better.
I didn't think I was a big fan of musicals, but seeing a broadway show is really an experience unlike any other. We had planned to go to two shows, but we enjoyed them so much that we ended up getting last minute tickets for another. We saw Mamma Mia, Curtains, and Chicago. Chicago was ok, but a little boring. Curtains was great and Mama Mia was so much fun. Like I said, I've never been a big "musical" guy and I'm not even close to being a metro, but Mamma Mia was fantastic.