I live in Stuttgart, and we don't get real winter here, -20 C happened once in the 8 years I've been here. It's mostly just unpleasantly cool around 3 - 10 C and damp and dark. Mostly just layer up, get a good rain shell and wear sweaters underneath. Wool winter jackets are more than adequate. Down jackets are WAY overkill here, I have one I brought over from Canada and haven't worn it once. Normal shoes are fine on 99% of the days here, but a sturdy set of hiking boots (real winter boots not at all necessary) will keep your feet warm and dry enough.
You'll notice the German kids all packaged up in full rain gear with toques, gloves and scarves from about Oct 1 onwards into spring, regardless of weather - I find this hilarious, it is so absolutely unnecessary.
You'll adjust fine, just make sure you take vitamin supplements, particularly vitamin D, the darkness more than anything will make you sick if you come from a warm country.
Welcome to Stuttgart, eventually you'll like it here - An acquaintance from Berlin put it well: Love at third sight.