Even though all the Corolla trim levels have approximately the same total tire diameter, the ones with larger wheel sizes will have a lower profile (aspect ratio) tires that are also slightly wider. The larger wheels with lower profile tires have better handling characteristics, and also different aesthetics (guys tend to like larger wheels), but they also have a slightly harsher ride and not as good in rain or snow.
I would go to Costco (if you are a member) or Discount Tire for winter tires. They will help you choose the right ones. Both of these companies offer free exchanges, so they are going to try and put you into a tire that you will be happy with. They both have excellent service, but best to go when they are not the busiest if possible. They both are national chains and have very good service. If the recommended ones are too expensive for you, ask them for other options. You can also look at the Discount Tire website for the choices available for your specific vehicle.
You need to decide if you want the winter tires mounted on your existing wheels, or whether you will buy new dedicated wheels for your winter tires. Not sure if Costco sells rims. Also, you may have to figure out how to store your regular tires while using the winter tires (and the reverse in the summer).