Ken's advice is good, but I took a different approach. I have to say, though, that my regular commuter is also my snow commuter, and it's a Surly Double Cross, a cyclecross frame with 700c tires. In the soft snow, the greater degree of weight on the front wheel was a little troubling and skittish at time. Although I had some fairly wide tires, that much weight forward led to a squirrely front end on the soft stuff, even more so when it was deep and fresh. As Ken noted, because I had studded snow tires, things went better on compact snow and ice.
My set up required that I take off my normal fenders, planet earth full coverage, as much as they provide full coverage. I also had to take off my rack, both because the tires were too tall and wide. I forget exactly what size my tires are, but they are studded snow tires, the sort of which are available from any bike store. They are at least 45 wide, maybe wider. The cyclecross frame allows for a pretty wide tire. Now that I mention it, I was able to keep the front fender on with the wider tires. It was the rear that just wouldn't fit. The rack either. What I ended up doing was taking a fender designed for a front suspension fork and attaching it to the cross bar in the seat stays where the regular fender would also attach. It worked well enough, keeping off most of the slop, but not all of it.
I also had to switch from panniers to a back pack. While I have the fenders back on for spring, summer and fall, I've not reattached the rack because I like the look of the bike without the rack and just use a backpack for commutes and errands.
For lights, I went two ways. One is a princetec fairly high power four-way light, with both LED and halogen. I don't think they make this model anymore, having replaced the halogen bulb with something else. This light is more about "Seeing" than being seen. I also have an LED cateye that flashes for being seen. I have no way to know how much more likely someone is to see me because of the flashing light, but I tend to run the steady and the flashing together. For the rear, I've read somewhere that during low light or dusky conditions, a steady taillight is a better, but during dark, the flashing is better. I tend to use the flashing all the time desite this. My flash has a sorta random setting that I use so it is less likely to entrance a driver. I've also heard of a study from Germany that indicated that drunk drivers are more "attracted" to a flashing light, attracted in a bad way, as in they steer into flashing lights. You may want to look into all this stuff I've heard because I'm not sure how valid any of it is.
For clothes, I have pearl izumi amfib tights to keep out the cold, smartwood socks and my regular bike shoes, Sidi something. I also have PI amfib shoe covers to keep my feet fairly warm and dry. When it was really cold, even though I had wool socks, shoes and the covers, I couldn't do much more than an hour of riding before my toes got too cold. I have a variety of long sleeved jerseys, polypro long underwear tops and a rain coat to block the wind. It worked pretty well in teh cold. I've also ridden with a softshell jacket that works pretty well, depending on the conditions.
Finally, cover your head, with either a skull cap that covers your ears, or a balalcava, not to be confused with baklava, which tastes better but doesn't keep your ears or head warm. For the coldest of days, I used ski goggles. Before going for the goggles, my glasses not only fogged up, but froze up, which limits visibility and made things a little dicier than I liked.
Winter cycling is almost like cross country skiing in how you have to dress. You should be cold at first. If you aren't, you are over dressed. Even then, you want to be able to open zippers and other vents to keep from sweating too much unless you'll be somewhere warm and dry fairly soon.