I mean tbh, I wouldn't bike in Manhattan. Most people are fine, but people do get in accidents doing it. I had a friend who was hit by a car while riding his bike in Manhattan when we were seniors in high school; he spent a few days in a coma before succumbing to his injuries. Since then the idea of biking here has terrified me...
But if you're adamant about it, there are Citibikes. I think membership is like 100 a year or something of the sort and there are two Citibike stations very close to school, one on E23rd and one on E25th.
Also though, remember that winters here can be awful. This winter from end of January to like mid-March we constantly had snow on the ground. At times, over a foot. And biking in that is pretty bad. Plus it was cold as crap all winter, so I imagine biking in that weather is less than pleasant. And it also rains a lot, hah.