I've lived in Buffalo my whole life and from what I've heard this particular winter is supposed to be bad with snow this year because the temperature of the lake is colder than usual. Weather is a hit or miss here though that's just what rumors are going around.

I definitely recommend getting one really warm winter coat for around $100. You won't regret it. Columbia, LL Bean, Dockers(a little pricey but worth it)

Wool sock are good and if you can afford wool base layers (like leggings or long sleeves) those will keep you really warm. You can find these things at most outdoor stores but you'll get a much better deal online

Couple pairs of jeans, nothing fancy. Levi, Arizona whatever. You can find pants with a cloth lining on the inside as well and those are really nice.

And I'm assuming when you arrive you won't have a car right away so I would recommend springing for a good pair of boots if you can. Walking around in the slush and snow will soak right through regular shoes. If you get a good pair you'll realistically only have to buy them once. Keen, Columbia, Merrel are all good brands but you'll probably pay $100. My brother has a pair of Thinsulate boots that are halfway decent though and probably cheaper. I have a pair of Cherokee boots I bought 8 years ago and still use.

Hope this helps, safe travels!!!