If travelling through excessively cold regions, what steps should you take to keep your vehicle warm enough to drive?

TL;DR Cold will kill you. There are things you can obtain in the cold place, and simple things you can do while driving, that might save your life.

Driving in the cold can be dangerous to the point of being deadly. I've been driving in Ontario (and living in the countryside) for over 35 years. I've been in the ditch countless times (usually some guys in a pickup truck come by and get me out, sometimes it's a tow truck thing), I've been delayed hours by road closures, I've cancelled plans because they say not to drive, I've slid and spun on all kinds of ice (fresh wet ice from freezing rain, old frozen stuff with soft white snow on top, black ice, and more) including sliding into things I didn't want to slide into, and faced hills that no-one could get up - you just have to turn around and go a different way. Wandering into a situation that scares locals is generally not going to be smart. However, if you find yourself in it, at least make the best of it.

You will need:

• knowledge of a local radio station to listen to for traffic and news. If the local news starts saying that the police are saying to stay off the roads, stay off the roads
• snow tires. Not "all season tires", snow tires. If you are driving only within a city and it is further than 10C or 20F below freezing then perhaps you are ok without snow tires. (A very cold dry road is safer than a slippery icy or snowy one.) Otherwise your "stay off the roads" filter will need to kick in sooner than everyone else's.
• warm clothes: hats, boots, mitts, serious coat etc. You don't have to wear them while you drive (your car surely has a heater) but you need them with you in case the car gets into trouble. Some people also bring a candle in a coffee can (and matches) and an extra blanket or two, but I've never needed these.
• snow scraper and brush, and some idea of how to use them. Rental cars are generally missing these (as will your car be if you drove from somewhere warm) but they are super cheap ($3, $5) so get someone to tell you where you can buy some and take a short detour to get them. (Get an extra bottle of windshield antifreeze while you're at it, in case you run out on the road, and by the way that place will probably sell you some ugly hats and gloves too.) Not being able to see out the windshield will kill you. Wear your gloves when clearing off the car and clear the outside mirrors, headlights, hood and roof as well as the glass. In a pinch a credit card can function as a scraper.
• food (a granola bar, say) and drink (a bottle of water) in case your one-hour trip turns into 6 or 7
• a working cell phone and a way to charge it
• knowledge of where you are so that if you call for help you can tell people where to find you. A map or a mapping app so you can work out detours if you can't just follow the crowd.
• a way to find gas stations, coffee shops etc, again for dealing with extended trips (last week a 2.5 hour trip took me 5 hours, using more gas, windshield fluid etc than I planned, and meaning we needed a bathroom-and-coffee stop along the way.) Install an app or do some searches before you leave. Learn to read the "what is at this exit" signs along the highway system you'll be using.
• understanding of the features of your car that you never use, or features in this car that your car does not have such as fog lights (also good in blowing snow), rear window defroster, the various "defrost, feet, faces" settings of the heater, etc. Learn how to fill the windshield antifreeze in the daylight in case you later need to do it in the dark.

About starting

If the car has a block heater, plug it in, even if that means parking somewhere less convenient. (The cord is very short: ask someone for an outdoor extension cord unless you're in Northern Ontario or Winnipeg where there are handy plugs at the parking spots.) When it's time to go, start the car, then get it all cleared off of snow and ice. This will get the oil and the engine happier and also get the inside a little warmer.

If you had a very hard time starting the car you may not want to turn it off for a short stop such as coffee and bathroom. If so, go in shifts so that one person stays with the running car. Cars are stolen in super cold weather because people leave them running and go inside. Note, you cannot leave the car running while getting gas.

Be gentle to your battery. Do not leave the key at "on" if the engine is not running: this will blow the heater fan, run the defroster wires in the rear window etc and drain the battery. And you need every speck of juice it has to start the car with thick sludgy oil. Use "acc" to hear the radio with the car off, or just leave it off. Also try to not use the dome light or headlights if the car is not running.

On the road

If it's very cold (say -25C) you may find the windows are icing up no matter what you do, and you may have to divert all the heating to defrost and none for your feet, or turn the fan up so loud that it is super uncomfortable. If it's freezing rain, you may find nothing will keep the outside windshield clear and you'll have to pull over and scrape. These are uncomfortable and inconvenient things but they are far better than waiting at the side of the road for police or ambulance after your accident.

Do not let the gas tank go below half full. You may need more fuel than you planned to go the distance, or if the crappy weather includes power failures you may not be able to get gas where you thought you could. Fill up more often for safety's sake. When getting gas, check the headlights and other lights - they may be coated with ice either if you're driving in freezing rain, or just because of mess splashed up from the cars in front of you. This can seriously reduce the light that gets out.

If it is slippery, slow down. Leave more space between you and the car ahead. Brake slowly - partly so you don't skid and partly so the car behind you has time to react.

Most important

Above all, don't be afraid to give up and just stay somewhere warm and safe, and head out again when the weather is better. Living somewhere very cold you understand that human wishes sometimes have to just wait until nature feels like letting it happen. Being prepared is safer than being unprepared, but sometimes the only safe thing to do is stay off the roads.