There really isnt a school for hunting, it is usually handed down experienc, and self taught. I would recommend taking a NY state hunter safety course. There are two, one for gun and one for bow. They are a wealth of information, are free, and will also potentially put you in contact with other hunters. The gun hunter safety is essential to get a hunting license in NY, the bow is neccessary for bow hunting BUT it has much more practical hunting information in it, including usually animal vital areas, blood trailing, and tree stand use. If you havent had much experience with firearms, I would also recommend a handgun safety course. The basic firearm handling practices are applicable to long arms, and safety can never be over emphasized, and you can never get too much firearm exposure. You can call around to local sporting goods shops (Gander Mtn, Dicks, local private gun shops) for classes being held.
I regularly hunt public land. NYS has millions of acres of it, and not all of it sees use at all. You do need a NYS hunting license. I do not know your situation for firearm possesion, so you need to clarify that. A call to the NYS Department of Environmental Conservation may answer that, or the NYS Police. Both groups are good people.
On hunting public land by yourself, you certainly need to put in your time scouting for proper locations. Not all woods hold deer (or rabbits, pheasants, etc.). A licensed guide (not all guides hunt- some mountain climb, fish, hike, canoe, or other sports) does not guarantee getting or seeing a deer. The animals have thier own plans, but they should guarantee a safe experience, and be able to offer alot of advice and learning. You can google for NYS guides. If you call one, (allways best to talk to several different ones) tell them your background and what you want to do/learn. Ask for references and call them up. Guides can and will use public lands they are familiar with.
A shooting preserve offers the opportunity to shoot different species on a private (high fence) piece of land, usually with the owner acting as a guide. success is usually 100%, prices vary greatly. For someone like myself who was raised hunting the big north woods, there is nothing to hold my interest. Alot of the hunting experience is the quiet time alone in the woods, sometimes for days on end with no one else around (not that I am not sociable, it is how I de-stress). A relatively small piece of land ( sometimes as small as 100 acres, or as large as several thousand) holding a known population of animals (certainly several times higher than occurs in the wild) well known to the owner removes the element of the hunt for me. For someone (like you ) new to the outdoors and hunting, it could be the opportunity to get a taste for what the experience is like in a safe manner. Nothing is wrong with that. As well, private shooting preserves do not have to follow the NYS hunting seasons, so it is possible to deer hunt (or elk hunt, or bison, if they have them) during different times of the year.
The game you start with depends on what you are most interested in. New York has alot of wild white tailed deer. Some guides specialize in small game- rabbits, grouse, turkeys or pheasants, in season. Small game hunting can be fast and exciting, with excellent table fare when you are done. Most shooting preserves offer Russian Boar hunting which is exciting and pretty economical, with great tasting pork when you are done.
It is great that you want to experience this, and I hope you are able to. For someone totally new to the sports, it can be a challenge. Sportsmen tend to be pretty generous, but also protective of thier sports and sporting opportunities. What part of the state are you going to be in?
Well, I also am throwing alot out at you here, hopefully it will be helpful.