Where are you planing on staying/ do you have a fake ID? I'm guessing no? There is so much to do I work/live down here and i feel like I've only seen 5% of the city at best.
^If youre looking for a view go to top of the rock (top of Rockefeller center), do that over the empire state building. Empire is open really late at night with is cool if you meet a chick and want to take her up.
Check out a baseball game citi is super east to get to via the & train, yankees is super easy to get to via the MTA since they re-did the lines. As a mets fan im partial but Citi is better IMO and you can get seems cheep for under 10 usually under 5 on stubhub.
youre going to have to do the touristy stuff grand central/times square...etc. The library by Bryant park is mind blowing
Go down to the village/soho/canal street
If you want a laugh go to BK williamsburg, park slope...
A bunch of cool places to shop/awesome stores
Eat a lot of food/street food is great here. IDK if there is anywhere else in the world with an many food carts. Within 3 blocks of my boj at lunch there could be 10 carts/trucks to eat at.
Not sure if you smoke or if youre going to be with your parents but go to some museums. Such a great way to spend a day MOMA, natural histury, met are the usually main 3 biggest ones.
Staten Island ferry is free which is a cool thing to do if youre down town. There are allways free concerts around, maybe check out a play on broadway-there are places to get 'cheep' tickets
you could take the LIRR to a bus to jones beach if you want to check of the beach for a day/
so much stuff to do. Just walk around and get lost-the city (manhattan) is super safe, where you will most likely bee so there's no need to be worried about getting lost and walking around.