I wear Red Wing work boots, but regular work boots won't be much warmer or dryer in the snow than your tennis shoes... This year was the first time since I was in grade-school that I bought snow-boots.
I bought "Irish Setter" by Red-Wing 1400gram insulated/waterproof hunting boots. They were around $110, money well-spent.

My feet have stayed warm & dry in even the coldest weather. Along with some good insulated bibs, gloves, and face mask, I've spent hours outside in -30deg weather without getting cold this winter.
Unfortunately, they are made in China...