http://www.dec.ny.gov/outdoor/45415.html
This map will let you know where you can hunt. There are other lands available as well but the state land is a decent place to start.
You have to take your hunter safety course and your bow hunters safety course. Passing the course only gives you a certificate that allows you to buy the license. From there, you have to actually buy the license. You buy according to what (and in a way, where) you will be hunting.
Westchester is bow only. Putnam County and Long Island is shotgun, muzzleloader, and bow but no rifles allowed.
It really helps if you have a car.
Here is a link to last years Hunting & Trapping Regulations Guide. I recommend you read it. This years edition isn't out till a couple of more weeks.
http://www.dec.ny.gov/outdoor/37136.html