I recommend you prepare for colder than -15C/5F
I lived in southern NY for a while, and I would like to suggest from experience that you prepare for colder than -15C/5F. While that may be the typical range, it is normal to have a handful of days colder than that scattered about the winter.
You should be prepared to deal with the occasional -20C/-5F to -23C/-10F days. And, if you venture up into the northern parts of the state, you should be prepared for the possibility of some days even colder than that. I remember encountering occasional temperatures below -29C/-20F.
The record cold temperatures for various cities in NY reach down into the -30s and -40s. The very coldest record for NY, from a quick Google search, appears to be in the -45C/-50F area in Old Forge... of course, you are not likely to encounter -50 if you are not doing much outdoor winter activity, but it is worth noting.
What to wear
Layers work well.
Depending on how cold it feels and what the weather forecast looks like for the day, put on one or more of the following. You can even pile it all on top of each other in many layers for the worst days...
The colder, the more layers. Thermal pants or sweatpants either instead of or over top of under-pants, and then jeans or work pants on over top of that. That is 2.5 layers over the lower half.
Larger, thicker socks over top of normal socks.
T-shirt, and a thicker thermal shirt, and a sweat shirt or work shirt, and a light winter jacket. If it is very windy or the coldest of days, I'll add a hoodie in there under the light jacket, and I will either replace the light jacket with a heavy winter jacket or wear the heavy jacket over the light one.
During the colder days, at least 1 pair of gloves. During the bad days, one of the small, thin pairs of gloves that you can pick up cheap from Wal-Mart, and larger, thicker gloves over top of that, and sometimes a big pair of mittens on over top of that.
If your exposure is limited, many people bring much less.
If you are just walking a few blocks away or if you are driving in a heated vehicle, you can get away with a lot less. Many people go out to start up their car, turn on the heater, and go back inside while their car warms up. Then they go out not properly dressed for the weather, because they will only be exposed to it for a minute or less.
If you forego too much because you expect not to be exposed for long, sometimes you might get stuck in the cold and be miserably cold. Sometimes the snow dumps quick while you're in somewhere, and you come out to a pile of snow that needs to be removed from your vehicle, and that can freeze you if you came underdressed because of your car heater.
Bring more than what you think you need.
You should keep an extra shirt or two, or a blanket, or something in your vehicle. Leave it there through the winter in case you need it when you did not expect to.
Sometimes the vehicle you are driving or riding in breaks down or encounters more snow than it can drive through or it just slides off the road; in any of those situations you get stranded in the cold for a while. Maybe 5 years ago or so the state throughway received so much snow that many vehicles on that highway got stuck for a while, a few days if I recall.
Personally, I keep 1 or 2 extra sets of clothes, a snow shovel, and a cold weather sleeping back in my vehicle just to be safe. Most of it has been used in winter emergencies multiple times.
Sometimes the visibility in NY gets to zero during the really bad snow storms; NY gets white-outs. When I say zero, I really mean zero. Sometimes you cannot tell where the side of the road is, you cannot read any signs, and you cannot even see the vehicle in front of you on the road unless its lights are on and you are less than 10 feet from it.
This severity of white out is not common, but most winters we get at least 1 of them. When this happens, if you are out in it, you need to make sure you are extremely careful, and if you are driving then drive extremely slowly. If you expect these conditions, it affects the clothing you should bring; bring everything I mentioned above in case you get stuck in it.
That all might sound excessive, but the point is just to make sure you have multiple layers available, at least 1 more than you feel like you need. When you get too cold, put more on. When you get too warm, take some off.
I also suggest you keep some food available in your vehicle, and I rotate bottles of water in and out of it to keep unfrozen water available. I have had to use both when stuck in the snow while driving the worst days in NY.
The winters in NY are much worse than most people farther south are used to, but they are easily tolerated with a little bit of preparation and save travelling.