You always need to turn on the headlights when you are driving, even during the daytime. Check out speed limits and penalties for driving too fast. Winter tires are not compulsory in Switzerland, but they are indispensable during winter in Graubünden. You should also keep a pair of snow chains in the trunk during the winter months, if you do not have a four-wheel drive. A four-wheel drive is an advantage in the mountains.
It is very common that you have to pay a fee for public parking in Switzerland, even in small villages. It is customary that you have to pay in advance in public parking lots and also in covered car parks. In many alpine villages, an overnight parking ban applies in public car parks during winter. In Davos, the monthly rent for a covered parking spot is around CHF 120.