It's basically a ski gear of some sorts, a base layer and a mid-layer you would need. Ski gear is fine on the streets also. You would blend in with the rest of us.
You can go very high-tech with your winter clothing, but I wouldn't over-exaggerate with your needs. It's not an expedition you going to but a vacation. Always near services and warm interiors. In my own experience, on every-day use, some cheap technical underwear is perfect as a base layer. Even cotton will do, if not doing anything too physical. We never use silk for example (other than as a balaclava). A normal sweater works as a mid-layer. And for the outer layer just some ski jacket and pants you are comfortable with. Don't worry about extra waterproofness as you probably won’t need it - except for kids, who you can't keep out of snow. But for adults I would say something like 95% of us don't choose gore-tex, for example, on every-day use.
Lined shoes with a thick sole are important. But again for a short trip no need for anything fancy, but your shoes should be roomy enough to comfortably fit a woolen sock, and you should still be able to wiggle your toes. A woolen scarf, a warm hat and mittens.
Normal winter clothing is simple, but it's a different sport if you are actually doing sports or going to safaris. But then on safaris they offer you with a good gear, a snowsuit, boots, gloves etc. You can also hire a snowsuit, but I bet you would get uncomfortable with it within a day.
If you are worried about your ski jacket being too thin, put on another layer first. If you are worried not packing enough layers with you, then don't. Weather might not be that cold anyway, and we still have shops here also. And I'm pretty sure up here our selection of winter gear is better than what you have down there... :)