There are three possible answers to this question.
1. Place to stay: New York City has thousands of hostels to choose from, everything from the YMCA/YWCA to ordinary hostels, some very well known. Two hostels that I know of are the American Youth Hostels, Inc, on the Upper West Side and the YMCA Vanderbilt Center in Midtown. I've included there web address at the bottom of this post, along with several other links to Hostels in Manhattan.
2. Transportation to and from New York: Since you're coming from The Windy City, my best bet will be take Amtrak into Penn Station, New York. They may have discounts for students.
Also Greyhound bus, which always offers discount to students serves New York City, with a stop at the Port Authority Bus Terminal.
Many low budget airlines travels between New York and Chicago, however they will land at airport far away from the city. You may want to try JetBlue, they have a hub in New York and JFK Airport. Check with them to see if they offer discounts to students.
3. Transportation while in New York: Since you going to be in New York there is no need to rent a car. NYC Transit, like Chicago's CTA, operates both the buses and subway systems. The fare on both the buses and subway are $2 a ride, but if you're in town for a week, you can buy a 7-days unlimited MetroCard, which is a fare card similar to the one in Chicago, for $24 and take as many rides as you want on the subway and bus.
I hope this information was very helpful. I need any travel direction while in town, just drop me a line at email@example.com.
Good luck and happy travels.
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