Do the usual touristy stuff, obviously: Empire State Building, Statue of Liberty, Metropolitan museum, MoMA, Whitney, Guggenheim, American museum of natural history...
but also spend some time just exploring different neighborhoods on foot: boutiques and record stores on the lower east side, catch a concert at one of the new venues in Williamsburg, brooklyn, hit a few bars in the East Village, spend some time exploring Central Park, etc, etc.
Quick Tips:
Try to get all of that Times Square/Midtown stuff out of the way in one day/afternoon and don't choose a hotel/hostel in that area if possible. But, if you are totally enchanted by the madness of midtown, at least try to be considerate to those of us who live/work there. If you're in a large group, don't walk side by side and take up the entire sidewalk. Admire the tall buildings, but also try to be aware of the cars and people surrounding you on the ground. Oh, and check out the theaters on 42nd btwn 6th + 7th ave. And avoid the subway at rush hour. It's misery.Best Way To Get Around:
You can get a day or a week pass for the subway/bus system. Even without the unlimited pass, you get free bus-to-subway or bus-to-bus transfers (*note: NOT subway to subway transfers once you exit the station). But New York is easy to tackle on foot. A good majority of Manhattan is laid out as an easy-to-navigate grid so it's hard to get lost walking at a moderate pace. I usually average about 1 number block a minute (avenue blocks are much longer), and there's so much to see that you'll hardly notice the time unless you're in a rush or the weather is bad.