Boot recommendations?

I need a pair of boots that are good for rain, mud, snow, and ice. I would mainly just be using them to take my dogs out, so I don't want to have to buy a pair of rain boots and a pair of snow boots. Nothing too warm, since I would wear them in the summer when it's raining. I can always size up a little and wear fluffy socks in the winter.

I don't even know what to be looking for. Any suggestions on a pair of boots or what to search? I can't spend much right now. Maybe like $60? We just had our first snow fall last night, which I didn't even realize was coming, so I'm on a bit of a time crunch.