DuckHunt. I am also new to hunting (started hunting 2 months ago) and I'm also from NYC so here's what you need to know.
#1. Gun license and hunting license are completely independent of each other. You can technically hunt without a gun license and you can own a gun and not hunt.
#2. You want to apply for your gun license as soon as you can. It takes about 10 to 12 weeks to process. You can forget about getting a handgun/pistol. In NYC, that's almost impossible. Rifles and shotguns are easier. You would have to go to the rifle and shotgun office in Queens. Here's the info. They open late on Mondays.
http://www.nyc.gov/html/nypd/html/permits/rifle_licensing_information.shtml
Technically, once you apply, you can purchase a shotgun or rifle. If stopped, police can check and find out that your license is in the processing stage and you would be fine but I wouldn't chance it though because they have to take your weapons into their possession until they can clarify that you have applied and who know how long that would take or how you would be treated during the process.
#3 Obtain your hunting license. Again, you don't need a gun license to apply for a hunting license. To apply for a hunting license, you need to complete the hunters safety course.
Here's a list of state-wide course schedule.
http://www.dec.ny.gov/outdoor/9191.html
If you don't see one near you or at a appropriate time, check back ever so often because they update it as new courses are offered.
Once you pass the test you will get your hunter's safety certification in the mail in a couple of business days. This allows you to go to any local retailer that handles hunting licenses and by a license. The hunters safety certification is good for life. Hunting licenses are only good for the hunting year so you will have to buy a new one each year. You can also purchase it online.
If you are going to trap and/or bow hunt, you need to take a trappers class and/or bow hunting class respectively.
#4 Rifle and shotgun purchase. New York State has the strictest gun laws in the country. New York City is even stricter. You have to remember that NYC seperates itself from the rest of the state. You can carry a shotgun without a license in Westchester but you can't in the Bronx. Know the law before you purchase.
Read the entire Penal Code 260 regulating possession of fireamrs.
.htm'>http://wings.buffalo.edu/law/bclc/web/NewYork/ny3(.htm
Rule of thumb. If you are not sure, ask. You can message me if you want. Second rule of thumb, if you go with a semi-auto, 20 million restrictions apply so tread carefully. If you stick with bolt-action, pump action, break-action, lever action, or single shot...you're much better off.
Hope this helps.