Simple answer: you can ride in winter in NY. If you can X-country ski or skate in winter there is no reason you can't bike.
But there are several things to keep in mind:
1) How often you ride will depend on whether you can dress for the conditions and how well the snow is handled on the roads you plan to ride on.
2) Cycling on snow-cleared roads on a sunny day is fun and relatively easy. Riding on snow more than a few inches deep is more tricky. I found that narrow tires work better than wide tires (they cut through better) so I'd advise a road bike rather than mountain. Studded tires can be a big help on ice. Despite the more slippery conditions, overall you are at less risk of hurting yourself in winter since you are probably wearing more clothing, going slower and more carefully, cars tend to give you more room (presumably they think you are mad and should be avoided), and if you do fall you are more likely to fall onto snow than pavement.
3) Dressing for winter cycling is very different to summer. Dressing in layers is advised ... and like with X-Country skiing, overheating/sweating is best avoided. You don't need heavy clothing on your body (although a windproof outer layer is mandatory), but hands, feet and face need protection. Plan on face protection, including goggles, at low (sub-zero Fahrenheit) temperatures. See the link for more info, or simply google "winter+cycling".
4) You should consider whether you want to risk your bike winter cycling. Salt on winter roads is bad for bikes. Paradoxically, better quality bikes stand up to winter better than budget bikes. But whatever you ride you will have buildup of "grunge" (see 3rd image at 2nd link) -- a mix of salt, sand, ice & snow -- that can clog narrow openings and interfere with gear operation. Use lots of oil to help protect from salt, plan on a mid winter overhaul, and to have to replace your chain (at a minimum) and overhaul the bike come Spring. Cheap bikes will literally seize up and die through winter cycling. You can be a fair weather winter cyclist and use a good bike without much risk, but if you plan to commute or ride in other than dry conditions you risk destroying your bike. That's why many winter cyclists ride a dedicated winter bike.
5) Buying a bike in the fall (or winter) can save a lot of money, so even if you don't ride it much in winter it can be a cost effective time to buy next summer's bike.
6) One approach to consider: mount your new bike on a wind trainer and ride to get in shape whenever you want indoors and, on nice dry sunny winter days take it out for a real ride.
As an aside, I commuted and rode for exercise year round for over 12 years in Ottawa (Canada) in temperatures as low as -35F, so I write from some experience.