For the car:
Keep a small snow shovel in your trunk. They make ones designed for this and you can pick them up in just about any hardware store and often online. Especially if you aren't used to driving in snow you might get stuck not knowing how deep of snow your car can handle. Having a shovel is far nicer than trying to dig your wheels out by hand in a snowstorm. Beyond that other folks have covered the basics: blanket, extra gloves, ice scraper, etc. We also keep a couple of extra supplies in our trunk: an extra bottle of windshield wiper fluid, some granola bars, and hand warmers that insert into your gloves (and if necessary boots/shoes):
https://www.amazon.com/HeatMax-Hot-Hands-Handwarmer-Pairs/dp/B089KP85VB
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Edit: As someone else said make sure your tires are in good shape. We use all-season tires, but we mostly just drive around the Twin Cities and try to avoid driving in storms. One other thing to be sure of is that your battery is in good shape. An old battery might be fine in warm weather but if it's more than 4-5 years old you might want to consider replacing it or risk your car not starting some day if it sits out in the cold for a while.
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Driving in the snow:
As someone else said, take it slow. Watch out especially if you are out driving during the first snow of the season. Half of the state seems to be incapable of remembering how to drive in the winter until they get used to it again. Often times there's multi-car pileups and cars in the ditches on the side of the road because people think that having 4wd means they can still drive 70mph in the snow like they do during summer. If you get stuck, try not to spin your wheels. You'll dig yourself in deeper and polish the snow making it even harder to get out. Instead slowly rock yourself forward/backward until you can get up the momentum to get back to shallower snow. You may need to shovel your wheels out before it will work. Watch out for drifts! They can be deeper than you might expect.
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Clothing:
One of the most important things to remember if you are going outside for the day is to layer. If you layer and get too warm you can always take something off, but if all you've got is a t-shirt and a huge down coat you are pretty much stuck with what you've got on. For boots, I love the North Face boots I bought a number of years ago. They are the standard rubber-lower leather-upper snow boots. There are tons of other brands that work too, but you'll definitely want a decent pair of water proof snow boots of some kind. The North Face, Sorel, Columbia, Ugg, etc all make decent ones for anywhere from $75-$250 depending insulation/style/etc. One thing to remember is that frostbite can happen quickly on extremely cold days if you have exposed skin. Especially if the temperature is significantly below zero, make sure you bring gloves, a winter hat, scarf, etc with you. Nothing is worse than trying to scrap ice off your windshield or pump gas in -20f when you've forgotten your gloves.
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Hope this helps!