I used to live in an area (PA) where you could conceivably have two sets of tires, but I never did. Always just bought a good all-season radial. Never had any issues. I did know a few folks who had snow tires, but for my generation, it seemed far from the norm. I think even my father didn't have specific snowtires on his car starting in the 80s.
The only type of tire I can see that would be of benefit in the snow is a studded tire, and in some places, they are outlawed or extremely limited in the amount of time you can have them on your car since they do a lot of damage to the road surfaces.
Get a set of all-season tires, also get a set of 'chains'...they are easy to put on if you have icy or heavy snow situations and will give you the extra traction in those emergency situations. Otherwise all-seaon tires will do you well...at least they had in my snow- and ice-driving experiences.