The Staten Island ferry is FREE (don't let anyone sell you a ticket) and you can see the docks and the statue of liberty. Just don't ... stay in Staten island. Come back.
The Metropolitan Museum is beautiful, and technically free. You can donate whatever you want for a ticket, just make sure you get the ticket at the desks inside. I, personally, rather you pay something because you are visiting and it helps it stay up. Just give em 5$.
I would say avoid the bronx, in my opinion... nothing really comes to mind to me for visitors to see besides the Bronx Zoo - an amazingly large zoo.
Avoid the far reaches north of Manhattan (above 96th Street on either the east or west) unless you like to get down and dirty with less touristy things and have experience with not being overwhelmed by things...
dont really feel like typing more on my own so ask something and I or others may see it.