I'm going to New York

From February 23rd to February 28th I will be travelling to New York with my family. 
I have never been in a city this big or in The USA in general. So I was just wondering if there's something I must see or do while I'm there. Seeing as no one in my family have been to New York before, we are doing the usual tourist stuff like going to the American Museum Of National History and The Empire State Building. I'm 16 years old so I won't be living the nightlife or anything like that

Now, there are some things I already have on my to do list. Such as; eating a hamburger/cheeseburger, going to a Taco Bell and visiting an arcade with a Street Fighter 4 cabinet in it. Almost everything "nerdy" is frowned upon in my family so everything that has anything to do with games or technology, I have to do myself (Yes, I know my family sounds a bit weird, but they just old fashioned). 
Any suggestions to what I can fill my time with that doesn't require much planning or being over 21 is appreciated.
PS: If you're wondering, I'm travelling from Norway.