Did they shovel? How did they get their carriages to move around? Did they understand that salt melted ice?
The funny thing is that we all know the answer to this but we just don't think about it much (I'll get to that later).
For the most part they just got around on the surface of the snow or they laboriously removed snow with shovels. Snow is actually not too bad to get around in on foot or with a horse, it's most problematic with automobiles.
Before automobiles, people weren't typically making trips longer than a few miles on a daily basis. You'd travel to your place of work, you'd travel to the market, you'd travel to shops, all of that could be done on foot with only maybe an hour or two at most of walking each way (roughly 6 miles distance). Walking speed in snow isn't significantly lower than on dry ground, so that's not hugely affected as long as there isn't a lot of ice.
Horses have longer legs than humans and are quite adept at getting through even deep snow, so riding a single horse isn't a big problem. But that leaves carts and carriages, which become more difficult in the snow. And that's where sleds and sleighs come in. Which would have seen extensive use during winter months in snowy areas.
Here's a video from 1898 of sleigh riding in Central Park New York. You can see that the snow isn't removed, it's just ridden on top of. You can also see that a wheeled carriage is also still being used in the snow. It wouldn't be as ideal as a sleigh but it would work ok.
And, of course, the memory of these practices persists culturally through to the present day in such elements as Santa's sleigh or the "one horse open sleigh" from the Christmas song "Jingle Bells" and so on.
As for salt, that wasn't used on roads until the 1940s, well after the advent of automobiles and snow plows.
Edit: to add some additional clarification. It's unusual for there to be continuous heavy snowfall throughout the entire winter, usually there are only a few instances of heavy snowfall with more instances of lighter snowfall throughout the season. Typically snow would be compacted along roads and walkways and then people would travel on compacted snow most of the time. When snow is compacted, as long as it's not too icy, it's pretty easy to walk on and ride a sled or sleigh across without impeding movement much. Deep, heavy snowfalls will impede movement initially since they require more work to trailblaze paths and in some cases might be so deep that it requires removal just to get into or out of houses and buildings.
If you will accept the east coast of Canada, the Jesuit Relations of the 17th and 18th centuries have a lot to say about the challenges and benefits faced by early European settlers--in this case, Jesuit missionaries--in the deep of winter in "these wretched lands." A theme that emerges, over and over, is that winter makes travel easier and being indoors harder.
Fr. Paul le Jeune, arriving in Quebec in 1632, describes learning how to walk with snowshoes from the local Montagnais--he was so sure he was going to fall on his face at first, with every step he took--but he had grown quite skilled (though not as good as the Montagnais).
In 1640, Fr. Joseph Marie Chaumonot wrote back to Rome explaining how, the previous winter, the heavy snows had actually aided the Jesuits in traveling to evangelize among the Hurons. Snow evened out the paths to travel on--but only, he writes, after it has had a few days to freeze over and harden. Before that, "one sinks in at every step." (The Jesuits here are traveling by foot, although they probably used snowshoes as well--Chaumonot tells us that among the Hurons, snowshoe-making is very specifically the women's task).
Meanwhile, the frozen rivers and lakes were especially helpful. Ice skates for transportation are actually an ancient invention, and were well known and used by late medieval Europe for both transportation and leisure. Chaumonot notes that, additionally, it is much easier to drag possession across the ice (on planks or sleds) than it is to carry them on mud/dirt road. He's not quite so happy with the bitter winter wind!
Meanwhile, the Jesuits found being indoors somewhat less pleasant than in the summer. For one thing, they had constant trouble with leaking and even collapsing roofs--their construction methods weren't up to the weight of New France snowfall. Additionally, their food supply was quite circumscribed in the winter--they weren't hunters, so while (writes Le Jeune) the Montagnais were chowing down on moose, the Jesuits ate dried eel (which, yes, they had known to eat--and maybe were getting from?--Native women).
But then there was the problem of actually leaving their houses. Le Jeune describes a real snowpocalypse:
There was everywhere four or five feet of snow, in some places, over ten, before our house, a mountain: the wind drifting it, and we, on the other hand, shoveling it away to make a little path before our door. It rose like a wall, all white, higher by one or two feet than the roof of our house.
So you can see that, yes, they shoveled. Elsewhere Le Jeune describes the Montagnais using their snowshoes to shovel, but I'm not sure whether the Jesuits adopted that practice.
The walls kept out the wind (and sometimes the roof kept out the wet), but early modern homes were built to stay dry rather than warm. Le Jeune observes that frequently, he had trouble writing because the ink froze in the inkpot. Even when he was sitting right next to the great big fire!
Physical exertion was a good way to warm up, though. As one unfortunate scholastic found out, it was also a good way to get a little too thirsty in an environment where you couldn't have liquid water with you:
One of our countrymen was thirsty, when in the woods, and so thought to lap a little snow from the axe which he held; when he touched the iron his tongue stuck fast, and froze so quickly and so solidly, that in suddenly withdrawing the axe, on account of the cold that he felt, he at the same time tore almost all the skin from his tongue.
Basically all of the Jesuit Relations are available for free online, in modern English translation, if you'd like to explore further!
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I think it's pretty important to pinpoint a time and location. I have read a lot of books about early New England. People traveled by horse and you didn't take your horses out in really deep storms. In less deep snow you'd take out sleighs. In weather like this you stayed in.
But remember- roads aren't like they are today. People traveled by water if they could. It was the preferred route by far.
There is a good book Midwifes tale about travel in Maine in the late 1700's. You'd more like travel by foot than anyway else in a deep snow.
In a big city if you had the money -horse teams would be doubled up in an attempt to take a carriage through the snow.
Rail service kept going it seems. If it was too heavy I am sure that stopped as well
Source - I just reviewed 500 pages of old newspapers with the key word blizzard