In general winter riding is similar to summer, just colder and you dress differently.
In my experience riding in snow is actually easier with narrow tires than with wider ones (they cut through better rather than float). That said, anything less than 2 inches or so is pretty easy to ride in.
The best bet is dressing in layers (and don't worry much about your body (a breathable jacket and rain pants is all you need) ... you won't get cold due to the exercise). Hands, feet and face are more important to protect.
Don't use clipless pedals if there is a risk of falling ... use platform style and boots.
If there is ice studded tires are useful. You may have to order specially and they aren't cheap (ca $50). I rode for years in winter without them, but I was in a city (Ottawa, Canada) that did a good job of plowing and salting.
I found a good bike stood up to winter salt, sand, etc better than a cheap bike. I rode beater bikes that lasted one winter, and a good road commuter bike for 5 years with minor repairs (e.g. new chain, disassemble and oil brakes) to refurbish each spring ... but the bike and components stood up well..
I'd advise fenders and good lights (the latter for visibility to others, not you since cities are usually bright enough to see without lights).
My 12 years of winter riding at temperatures down to -35C will I'm sure make NYC seem tame by comparison ... most of the time. But don't be reluctant to take alternate transport on really bad days. It's not worth the risks to ride in 4"-5" of snow!
Hope that helps.