Prepare for the trip
Many road passes close for winter. There are no fixed dates as snow fall determines how long the roads are inaccessible. It's not uncommon for the higher passes to be closed until June. Alternatives to closed passes are car trains, a detour, or travel by train. Some passes are too narrow or steep for trailers and mobile homes, and for some it's not recommended to take along a trailer or mobile home. Make sure your car is in excellent condition. Also make sure that you have enough gas to make it to the next village, as there are no gas stations on pass roads.
You can use Google Maps to find routes and traveling times. Keep in mind that the indicated time is without breaks, and traveling on other roads than the motorways may take more time than estimated.
Prevent an overheated engine and mind the brakes
It is important to utilize the hauling capacity of the engine when ascending. This means that you do not change gears as quickly as on flat roads, in order to keep a high engine speed (3000 - 4000 revolutions/min). The speed of the vehicle should be relatively low for safety reasons. Assuming a 5-gear car, it is usually sufficient to use gear 1 to 3 only.
You should also use a low gear and a high engine speed when descending. You can often use the same gear that was useful during the ascent. The deceleration will (partly) be done by the engine this way. It saves the brakes, that can get too warm if you use them continuously. When you continue to gain speed, pick a lower gear or occasionally gently use the brakes. The accelerator pedal will not be necessary during most of the descent. Do certainly not drive down with the transmission in neutral. In most cars with automatic transmission, you can just use the regular D-position. On slight grades, it may be necessary to pick a low gear to prevent the car from changing gears too frequently. In such cases, use position 1 or 2 or the mountain gear if available.
You need to pull over if your car indicates that the engine or brakes are too hot. An unpleasant smell may be an indication as well. If this happens you have probably used an incorrect gear. You may turn on the heating to withdraw energy from the engine. Turning off the air conditioning might help when ascending as all energy remains available for the engine now.
Do read the manual of your car.
Adjust your driving style
Driving in the mountains is all about safety. It requires an adjusted driving style. Mountain roads can be narrow, have a lot of sharp curves and may pass along deep ravines. A crash barrier or fence, which is mostly present, is not always designed to actually stop a vehicle. Also, you'll be sharing the road with cyclists, pedestrians and cattle. Not all the time, but you need to be aware of it. So, concentrated driving matters greatly.
Your view can be blocked heavily in curves with rock walls or trees along the road, so you need to adjust your speed there. Sometimes there is a mirror to see around the corner. You need to decelerate before the curve, not when you are already in the curve. Once in the curve, you may step on the gas to improve traction. Be sure to keep on your own (right) lane, especially in outside curves. Some people tend to steer towards the inside of the curve, which can cause serious accidents with oncoming traffic.
Obstructions on the road
Not only cars are using mountain roads. Beware of cyclists, hikers, cattle and rubble. Especially be aware of cyclists in tunnels. Overtaking descending cyclists is usually not necessary since they are at least as fast as cars. Keep at a safe distance though. Drivers of slow vehicles may help you with a short signal with their right indicator to show you that you can overtake safely. You can do that yourself as well if faster drivers are following you. You can warn following traffic by switching on the emergency signals. Do not pull over on places with rubble on the road, because it indicates a risk of more rubble coming down.
On narrow roads where one can't see far ahead, you may want to warn other traffic that you're approaching. You may honk during day time or use light signals during night time. Excessive honking and honking in towns is not allowed.
Ascending traffic has priority over descending traffic, because descending vehicles can continue more easily after coming to a full stop. So, on narrow roads, descending traffic has to clear space for ascending traffic. A postcar always has priority, even if it is descending while you are ascending. When curves are ahead, the bus may announce its presence with its characteristic horn. Stop well before the curve because a bus in a narrow curve needs all available space. Keep plenty of clear space when overtaking ascending cyclists, and use your direction indicator.
Mountain roads may have unlit tunnels. Double check if your lights are switched on, take off your sunglasses and adjust your speed. Tunnels are often cold and moist so water may drop down.
Parking on a slope
You can do a number of things to park safely on a slope. You definitely need to use the handbrake and shift the transmission to first gear. When the slope is very steep you may put a stone behind the wheels and turn the steering wheel so that the car cannot roll very far.
What to do if the brakes fail
Acting quickly is necessary if you notice that your brakes fail. Try to decelerate by pulling and releasing the brakes repeatedly, or try one more pull with hard pedal pressure. Only pull the handbrake if the road is not slippery, otherwise you will lose control of the vehicle completely. If necessary, change to gear 1 or 2 without shifting to neutral, hopefully causing the engine to decelerate. You can also turn into an ascending side-road or meadow to decelerate, or scrape the car at a small angle to a (rock) wall along the road.