Read the previous article on Tips for road trips. Everything that applies to fair weather, applies even more so to inclement weather.
Warm clothing (including ski caps, gloves, and rubber boots that fit over shoes) and blankets (or sleeping bags). If you live in a mild climate and can't find such items where you normally shop, try a sporting goods store. In larger cities there are outdoor sports shops that carry all year round clothes to ware in very cold conditions.
Extra non-perishable ready-to-eat food
First aid kit
Extra prescription medication (if applicable), especially medication that can mean the difference between life and death in the short term, such as insulin for diabetes, inhalers/EpiPen for asthma/anaphylactic shock, (cortico)steroid treatments, etc.
Lighter and strike-anywhere matches
Portable radio with AM band and batteries, even if your vehicle has one.
Flashlight (torch) with batteries
Ice scraper with a brush.
A lightweight snow shovel (or any shovel, if your can't buy a snow shovel locally)
Sun glasses -- on a sunny, winter day, the glare on the road from snow (or even water) can be quite bad
If your vehicle doesn't have a navigation system, consider adding a USB GPS receiver w/software to your laptop computer (i.e. Microsoft Streets and Trips, Delorme Street Atlas, etc.). Hand-held units are also available if you don't own a laptop. One wrong turn onto a winter-abandoned road could be fatal.
If using a laptop, you'll also need a 150-watt or higher power inverter (converts car battery to household power mains voltage) or a laptop car adapter with the correct DC voltage and plug. Usually the inverter is much cheaper, and can be used for other low-wattage appliances (except clocks).
Don't forget your cell phone and its in-vehicle charger. However, service may not be available in rural areas.
Spare vehicle key to carry on your person
Bucket of sand in case your vehicle gets stuck (especially without all-wheel drive)
Tire chains or cables (see below)
Last, but not least, spare batteries for all the above battery-operated equipment.
If at all possible, allow for a couple days of flexibility so you can wait out a winter storm (preferably before leaving home). This is especially important for those who have little winter driving experience. Check the forecast before departing and try to avoid the worst of it. The most popular sites online are weather.com and wunderground.com.
Check highway conditions for all roads you will be traveling at the state/provincial highway department's website (search on Google, Yahoo, etc.). In the USA, include the word "state" (i.e. "Montana state") in your search.
If using mapping software to plan your route, always double-check it yourself against a detailed, printed map or atlas (e.g. Rand McNally, AAA, Michelin, etc.) Select "fastest" (not "shortest") routes, as this will help keep it to the main highways. Avoid anything below a state/provincial route except for the last few miles/kilometers to your destination, where necessary. Short cuts in winter can be deadly! On some minor roads in wintertime, there might not be another vehicle for days, weeks, or even months. Software maps are not as good as paper maps in showing how "minor" a less-used road is. Some aren't even paved, but look as good on a computer screen as a well-paved, busy thoroughfare.
Always get advice from locals beforehand about conditions off the main highways. The best route may not be obvious to you, or even be shown on a paper map. Also, let them know when you plan to arrive, with updates on any delays.
A four-wheel-drive vehicle is preferable to a two wheel drive.
If 4x4 vehicles are unavailable, a vehicle with a limited slip differential and/or traction control is highly recommended for driving on snow and ice. Two-wheel-drive vehicles without such a device can very quickly get stuck. In this situation, one wheel will block (have no power), while the other spins freely, unable to move the vehicle from this spot. An ordinary differential (most vehicles) provides equal power to the left and right wheels. Thus, if one wheel is on ice or buried in snow, it takes almost no power for this wheel to spin, while the other wheel receives likewise, but needs a whole lot more.
Front-wheel-drive is preferable to rear-wheel-drive (most Mercedes and BMW). Also, most smaller and mid-sized cars have front-wheel-drive nowadays.
A small, lightweight 4x4 vehicle with good ground clearance is preferable to a SUV or off-roader, especially on mountain roads. Narrow tires perform better on snow than wide tires. Heavy vehicles are more likely to slip on steep roads, and are also more difficult to rescue by pushing.
The same rule applies to two-wheel-drive vehicles: A subcompact is better for snowy conditions than a mid-size or luxury car.
All cars manufactured in the past few years for the USA, Canada and Europe have ABS. Elsewhere, ABS may be a bonus, so ask for it.
Preparing your vehicle
Check anti-freeze (coolant) levels, and have it changed every two to four years (depending on type used).
Make sure brakes and ABS system are in good condition.
Check heater and defroster and make sure they heat with no smell.
Check condition of the tires and their pressure. Don't forget the spare tire and jack.
A vehicle's battery can't work as well in cold weather. Have it replaced if it's near or over the pro-rated warranty period. On an older vehicle, check the belts, and an alternator test is also a good idea. Fill an older, unsealed battery with distilled water (but don't overfill).
Replace worn wiper blades and top off washer fluid. Don't use plain water as it will freeze. On the other hand, antifreeze intended for the radiator may damage your vehicle's paint. Look for the winter-type washer fluid at auto parts stores which has a freezing point of about -20F/-30C. Even though it's on the other side of the windshield, using the defroster also helps prevent the washer fluid from freezing.
Studded snow tires are good for winter driving conditions, though a few U.S. states prohibit them (except, perhaps, for out-of-state vehicles just passing though). On the other hand, in Sweden it's illegal to drive without proper winter tires in the winter, they don't have to be studded though. The tread depth must be at least 6/32" or 5 mm (or whatever local law requires), which is several times deeper than for regular tires wearing out. If you live in an area where it doesn't normally snow, it's probably not worth the time and expense.
Especially without snow tires, always bring tire chains or cable chains. Tire chains give better traction, but are more difficult to install and remove. Know your tire size (e.g. P195/60R-15) before purchasing. When needed, install on the drive wheels (i.e. front for front-wheel-drive, rear for rear-wheel-drive). If unsure about drive wheels, all rear-wheel-drive vehicles have a black sphere-shaped thing (the differential) on the axle between the rear tires. Note that 4WD/All-WD vehicles will have one there also. For 4WD/All-WD usually the front is best, but check owner's manual. Only use chains in snow or icy conditions, and remove them as soon as they're no longer needed. Don't even try them on for size on a hard, bare surface such as concrete. They might spin out and damage the chains, concrete, and/or wheel well of the vehicle, and possibly injure someone.
On the road
Driving on snow, and especially ice, requires extra stopping distance. Use extreme caution when going downhill.
You cannot drive at highway/freeway speeds with chains on.
Keep your fuel tank at least half full at all times.
For vehicles with ABS anti-locking brakes, do not pump the brake pedal if you start to skid.
If you miss your freeway exit, get off at the next one and turn around. Continuing on to a less-used alternate route in winter is foolish.
If a mountain pass is closed due to the weather, there's a good reason for it. Don't even think of using local roads to get around the closure. Saving a day or two is not worth risking your life.
In blizzard/white-out conditions, you may not be able to see anything though the windshield. Try rolling down the window and sticking your head outside. Then, find a safe place to get off the road and stay there until conditions improve.
Ice is more likely to form on bridges. Slow down when going over them, especially on the motorway.
When driving downhill, no amount of technical gadgetry (ABS,4x4,ESP,...) will protect you against skidding. Snow chains are the only remedy.
In case of emergency
Stay in your vehicle. It can provide enough shelter to save your live. Also, it's much easier for rescuers to see.
Run the engine for only 5-10 minutes each hour, with the heater on to the max. Make sure drifting snow doesn't block the exhaust pipe. Check each time before restarting engine (unless obvious not snowing/no wind), and shovel any snow out from the rear end as needed.
If you must go outside to search for help or food, do so only in the morning hours after sunrise when it's not snowing or foggy, and then only for one hour each way. If for some reason, you've left your vehicle and can't return, make or dig a snow shelter (i.e. primitive igloo) at once.
Assuming no cell phone service, turn your phone on and (totally) off every 15 minutes per hour to help save its battery. But don't waste the battery trying to dial numbers where there's no service. Rescuers can use portable receivers and direction finders to pick up its signal. However, even if they do, it's not possible to communicate with you over the phone.