As far as vehicles go, the best ones are those that maximize the amount of weight on the driving wheels. The worst will be a rear-wheel-drive that is nose heavy. A two-wheel-drive pickup truck is the best example. A rear-wheel-drive vehicle with an even weight distribution will be significantly better. Front-wheel-drive is better yet because typically 60% of the weight is carried by the front wheels. (Old, rear-engine, VW beetles and Chevy Corvairs also did well because 60% of their weight was on the rear wheels.) Best of all are all-wheel-drive vehicles because 100% of their weight rests on driving wheels.
There are summer tires, winter tires and all-season tires. Summer tires are hopeless in even a fraction of an inch of snow. Winter tires such as Bridgestone Blizzaks are the best for driving in snow. All-season tires are a compromise, better than summer tires but still significantly inferior to winter tires.
When the snow gets really deep, you need lots of ground clearance. Once you become “high centered”, everything else is irrelevant.
Unless you have to drive frequently in several inches of snow, your Corolla should be adequate. However, you will almost certainly need something better than all-season tires. (I have seen too many front-wheel-drive vehicles with all-season tires spinning their wheels helplessly in a couple of inches of snow on a gentle slope.) The best bet would be an extra set of wheels with good winter tires. Switch to the winter tires at the beginning of winter and go back to the all season tires in the spring.
The next step up is a small, all-wheel-drive vehicle. Examples are Subaru sedans and wagons, Subaru Forester, Honda CRV and Toyota RAV4. (For several years, my winter beater was a Toyota Tercel wagon that could be switched from front-wheel-drive to all-wheel-drive with a lever on the center console. Unfortunately, that model has been out of production for so long that you are unlikely to find one in decent condition.) With all-wheel-drive, you may be able to get away with all-season tires. However, if you have steep hills to climb, you should still get the extra set of wheels and winter tires.
The ultimate is a full size, all-wheel-drive pickup or SUV because of the increased ground clearance. If you need the extra clearance, you also need winter tires.
The down side of all-wheel-drive and/or full size vehicles is that they burn more fuel than your Corolla.
Finally, no matter what your level of preparation, it can simply get too deep for you. When it does, stay home! You won’t get to your destination anyway and you will be better off at home than stuck in the middle of the road part way up a hill.