Generally you will need a 4 Season Tent. https://www.rei.com/c/backpacking-tents/f/se-4-season
Mostly these are made for backpacking rather than family "camping" as such. These tents are made of heavier materials, are designed to shed snow and are usually single walled.
You can use a 3 season tent in winter in good conditions when snow is not expected. Tents with a steep pitch where the fly comes down to the ground will provide better weather protection.
I agree that you should look at a four season tent. There are quite a few available, and the one which with I have the most experience is the north Face VE-25. I first occupied this tent on a trip to Denali in 1987, where it withstood 80 mph winds quite handily. A few year later, i purchased four VE-25s for a long term project at Channel Islands National Park. - no snow but occasional high winds (the highest we experienced during the project was 60 mph). They worked very well for a job that lasted four three month seasons
I really like this tent. A lot of space for the packed weight, well made to stand up to challenging conditions. I am sure it is not the only such tent, but, given my experience, I would recommend this model without hesitation. The fact that this model has been around for more than thirty years says something about its suitability. I understand the current version has a second entrance (nice!).