Surfcast;
I am from NY. You probably have shot at the Calverton range like I one did. I am over in the catskills now.
You mentioned a high powered rifle and a family too. I have a basic requirement, the firearm must be capable of being used by all family members.
I'd get that 22lr sooner than later. You want to get your family more interested in shooting. Handing your wife a .30-06 rifle right off the bat is an excellent way to scare someone from shooting a gun. As far as dangerous animals in NY state, there is only one you should be weary of, deer ticks. Next to that, there isn't much next to that.
I am not saying a long range rifle is a bad thing at all. However, most of NY state is rather wooded. Something that can shoot 200 yards is plenty in NY. Guys who harvest deer every year may get one past 100 yards in NY. Some don't even get that far of a shot in their life. When deer are pressured they do not wander out in open areas during daylight. You best bet at a consistent meal is small game anyway.
The next purchase I think should be a 12 or 20 gauge shotgun. 12 gauge guns weigh more, so it offsets the recoil a bit. Careful selection of 12 gauge loads can be instrumental in having a low recoil gun. I prefer semi-auto shotguns. Despite what some folks will try to say online, they are very reliable. I shoot about 75-150 shots a week at the trap shooting range with a newer 11-87 remington 12 gauge. In the two years I have owned it, not one single failure. IT also has less recoil than my 870 in 20 gauge. Shotguns are very effective at every kind of hunting offered in NY. Shotguns are also one of the best self defense firearms out there. Law enforcement to some extent can pick what they want for a long arm, almost all of them go for a shotgun. Every gun can kill, but few can stop a threat right in their tracks. Threats to life rarely happen past 15 yards, very rarely.
Back to the rifle. .30-06 is fine for you, but might not be for the wife. A marlin 336 in .30-30 with a rubber ended buttstock is fine. Some women fine the metal end of the marlin's stock rough on them. It focuses the recoil force at times right on the lower edge of the stock. A good part of the problem is improper form. Buying a stock with recoil pad solves this problem. However, adding one on there just makes the length of pull too long for most women. Most women are already a bit short for the regular length, and then the recoil pad adds another 3/4 inch.
Semi-automatic rifles with a capacity of 5 rounds or less is legal to hunt with in NY. An AK-47 or SKS would make an excellent low recoil rifle and would be fine for hunting in virtually all public hunting spots. Be prepared to spend a bit of money to scope it. semi-autos are more brutal on scopes and a solid scope mount costs a bit of money. Some guys hunt open fields, but not many. Best bet is to hunt 50 yards in from the field, or a small wooded spot that connects or intersects several fields. But, public hunting is all wooded.
7mm08 and .243 are popular rounds that offer low recoil to shooters. .308 is easier on the shoulder than .30-06. You can also buy managed recoil rounds. They generally are lighter bullet weights with similar velocities as the standrd bullet weight rounds. For the most part, they do not effect your point of aim at 100 yards. Remington sells them. You can also reload your own ammunition mildy, so your family will not mind shooting the rifle.
NY is in no way a place that puts a high demand on a hunting rifle. Most game is easy to kill, just hard to find. Ranges are very close compared to other states.
I'd say get a 22lr, then a shotgun, and then a rifle. Also, if you are getting into hunting, start off with small game only. Learn the woods a bit, learn where to hunt, what to look for while your out there, and enjoy the outdoors. Hunting styles are so varied. You also want some good luck to run your way. Small game affords much more opportunity to pull the trigger than deer.
I am assuming you live in Long Island. Suffolk county is shotgun only. Going North in NY, Dutchess county is shotgun only. I am not sure what rockland and Orange counties are, I wouldn't be surprised if it was shotgun only.
The best one firearm to own if srictly talking big game hunting is a muzzleloader. You should be out there practicing, atleast once a month. If you do, and you practice just like your going to hunt, like you should; you only need one shot. Modern muzzleloaders are more accurate than shotguns. They're are also a 1/3 of the price to shoot. They're an acceptable alternative to shotguns in shotgun only areas in NY. The big thing though, as far as firearms hunting goes; the best hunting is muzzleloading season, usualy 10 day or so long and ends right before christmas, I see more deer out in muzzleloading season than I do the whole rifle season. The deer calm down, also the deer can not miss a day of eating because it is colder and the food available to them is not as nutricious. I love my thomspon center omega, but for about $200 every traditions muzzleloader I have seen has been a good shooter. Keep it clean and change the powder after a rainy day and they're 100% reliable. I have tried to make mine fail to fire, never got there.
Far as hunting goes in NY and in particular an area close to where you live, nothing beats hunting with a bow. If you live near levittown, cedar creek park has a no charge archery range. I see far more deer during bow season than with a firearm. Sometimes I see more opening weekend with a bow than I do the entire rifle and muzzleloading season. There are many reasons why. Also, the only private land I am allow to hunt is only because I hunt bow there.
Feel free to PM me if you have any NY specific questions.