Some great tips here but I'd just reinforce/add:
1. fenders, fenders, fenders!!!!! Long front one- even if it clips your toes a bit on turns- you'll learn to deal with it and keep your feet out of the way. Rear should go all the way to the rear stay if possible and below- it keeps grime away from your crank and bottom bracket as much as possible.
2. If you are in a relatively flat area, switch to a single speed. You can run a single cog front and rear and no longer need to worry about the derailleurs or their cables. You can also run a bmx chain- more durable and much cheaper to replace. And when you finally kill your chainring and cog, you only have to replace one! I thought it would be a pain, but honestly your top speed becomes limited in snow and ice.
3. Surprisingly if you live somewhere with salt, it's often better not to wash your bike when it is freezing out. Salt does all it's worst work when it is in solution- if it is below freezing not much happens. Wet it down without fully washing it off and it just gets salt water into crevices and makes it worse. Bringing it from cold to warm areas is also a bad idea in general. It encourages condensation and salt water to stay active and penetrate into places it shouldn't go more effectively. I generally only washed my commuter in winter if it was going to be warm for a while.
After doing this for years, I learned that winter commuting is generally hard on equipment, so just learn to live with the fact that things will have to be replaced over time. Single speed cogs generally lasted a year, and I replaced them every spring, or swapped back to multi-speed if I felt like it. Wheels lasted a few years. And after year 1, all this happened on my dedicated cheap commuter- I have other decent bikes for other rides. Being a certified bike mechanic helps!