Staten Island Ferry is A MUST!!! It's free, its 20 minutes each way, it is a 'cooling' experience after all the heat and humidity if you sit outside. Just avoid rush hour, as it IS A commuter vehicle. Inside the Staten Island side terminal there is a cool fish tank. Not much to do on SI near the ferry-- but there is a wonderful zoo on the island, especially if you are into reptiles/snakes.
http://www.statenislandzoo.org/
Snug Harbor Cultural Center is relatively near the ferry, they have a lovely Chinese Scholars Garden
http://www.snug-harbor.org/
Near the ferry in Battery Park is where they put the Sphere on display after 9/11,
http://en.wikipedia.org/wiki/The_Sphere
and the Bull on Broadway is nearby,
http://www.flickr.com/photos/flissphil/2295723628/
as is the The American Indian Museum.
http://www.nmai.si.edu/
Ground Zero is a few short blocks away.
If you go to the City Quilter, look into the Highline Park, nearby. A new and unique park.
http://www.thehighline.org/about/maps
And of course, the Met.
http://www.metmuseum.org/
The other responses have given you great ideas too!! There is way too much to do and not enough time.
GOOD shoes. Comfortable shoes.
Buy a Fun Pass for the subway, you walk or take it or a bus (slower!!!) everywhere, although a cab ride in NYC is probably an experience you shouldn't miss!! Have FUN. New York is a FRIENDLY place, and SAFE one if you just use your head!!!