I cycle through the winter, too. It's great being the only person out on the road on a sharp, clear morning.
Anyway. Here's how I do it:
Bike - The first thing you must do is fit mudguards. Riding a bike through slush and grit without mudguards is one of the most unpleasant things you can do.
I ride a road bike with a good-quality 1970s Reynolds-531 steel touring frame - it's supremely comfortable and has enough clearance for mudguards and wide tyres. Pure racing bikes will only accept narrow tyres and no mudguards at all, so I would avoid them. Mountain bikes make a lot of sense in winter for this reason (I'm thinking of buying a 29er mountain bike for doing more offroad, especially in winter).
The bike is set up with a double-sided rear hub (so singlespeed on one side and fixed-gear on the other). In winter I put bigger rear sprockets on to bring the ratios down to about 60" - this is very low, but in winter I don't mind travelling a bit slower, plus I go off-road more often.
Fixed gear is good in snow/ice because you can feel the rear slipping through the pedals and control it much more easily.
Read more about fixed-gear.
You can get studded tyres for impacted ice - two companies that make them are Nokian and Schwalbe (the tyre is the Marathon Winter, which is a plain old Marathon with a load of studs in it). I've not had to use studded tyres yet, although I did have to get off and walk occasionally last winter, so I might buy some.
The main tyres I use are: Vittoria Randonneur 28c, which are amazingly good in the wet and last a long time.
Kenda Smallblock 8 35c cyclocross tyre - I use this for travelling on dirt roads/gravel/over smoothish fields. It's got good grip off-road (slower on-road) but it's not as hardwearing or puncture-resistant as the Randonneurs.
To clean my bike and chain I use oil-free baby-wipes. I use these on everything, really. They work very well. Winter gets more grit onto your rims and chain so it's important to clean them after every ride. Cleaning a chain on a singlespeed bike is much easier than on a geared bike. For chain-lube I don't bother buying expensive shit from the bike shop - I use EP90 gearbox oil (NOT ATF fluid!) which is about £6 for a litre bottle. It works very well all year round. Wipe off the excess to stop dirt sticking to it.
As for clothing, I wear merino base layers, some full-length fitted trousers, and a Madison windproof/showerproof hi-viz jacket. Sometimes if it's really cold, I wear a long-sleeve jersey too, but the UK isn't very cold (just very wet). A merino skullcap and neck-puff-thing are both very useful. Two pairs of merino socks keep my feet warm - incidentally, since your feet are now larger you need to loosen your shoelaces up a bit, otherwise you'll restrict bloodflow to your feet and they'll get cold. I wear some pretty basic Goretex gloves with merino liners if it gets too cold - remember that brake levers can be harder to operate when wearing big gloves.
Lights are obviously pretty important in winter - contrary to popular belief, the front light is by far the most important; if there is a junction in the road and you have a dim headlight, drivers will pull out and you'll go over their bonnet - drivers overtaking you have a much longer time period in which to spot you. I use the well-regarded rechargeable Cateye Single Shot (which was about £60) and a USB-rechargeable rear light. If you travel on unlit roads or off-road at night, a good front light is necessary for spotting obstructions.
I also have some backup lights called Reelights - these have magnets mounted on the spokes and induction coils in the lights; when the magnets go past the coils it induces a current and the lights flash, giving you a small flashing backup light whenever the bicycle is in motion (and for a few minutes after stopping, too).
I think that's about it. Remember that you'll generally have less energy in winter (because until you warm up, you have to burn loads of energy to keep warm (and also possibly a small aerodynamic disadvantage due to cold air being denser)) so keep some flapjacks in your pockets to keep you going. If you can fit a thermos flask on your bike then that's great - if not, then you can wrap up a normal water bottle in a few thick socks and put a plastic bag over it - this keeps hot drinks warm for a reasonably long time.
Incidentally you can use the same trick for keeping drinks cold in summer - put a sock over your drinks bottle and wet it; as you cycle, the evaporation of the water in the sock will cool the drink. It works remarkably well.