When it comes to bike setup for winter riding, there are lots of things you can do, many things you should do and a couple of things that you pretty much have to do. I'll try to go through that list in reverse order (based on my opinions).
First and foremost, you need to get lights, at least two. Let me rephrase that, you need a minimum of two lights. You need a red blink light pointing when you've been, and some sort of headlight pointing where you are going. All bicycle manufacturers put reflectors on bikes as a requirement. I recommend you immediately remove the reflectors that came on your bike and replace them, with lights. Reflectors only work when light is pointed at them, like car headlights. When someone is pointing their car at you, generally you are in a bad position. Lights however, allow you to be seen even if someone doesn't have their vehicle pointed at you. And believe me when I say this is the preferable situation (yes, I have been hit). Riding without lights screams DUI cyclist. In which case many people will try to take you out before you get your license back and begin to endanger people again with your alcohol induced swerves and wanderings. Don't let this happen to you. For a taillight I recommend the Super Flash by Planet Bike. It really is awesome. Princeton Tec also makes a light called the Swerve which is similar in design and principle. While regular blinky lights are great, these two wonderlights are both incredible bright and have lights that flash at different intervals. These different intervals help someone seeing the light correctly place depth perception. That is to say, not only can they see you, but they can correctly place how far away you are. Both of them are also great on batteries. I change the batteries on my Super Flashes once a season. As for headlights, I recommend whatever you can afford. Cheap headlights ($50 or less) will allow you to be seen, which is the important thing. However, you have to spend more (around $80 to $90 dollars) to get lights that you can actually see with. Lesser lights put out light that you can quickly outride when on a bike, so you shouldn't count on them if your commute path or trail has totally unlit sections. In this secondary range my first pick is the Apex, also by Pinceton Tec. It is a legitimate backup to a HID or superbright LED system. Its major drawbacks are that the batteries are not rechargeable and are exposed (except for the Extreme version of the light). In the expensive but worth it category, I'll list Nite Rider. They have a full line of LED and HID headlights. I think everything they make is rechargeable, their stuff is bombproof (also cold proof, which is of note for cords) and if you do manage to break anything their customer service and repairs are the best in the industry in my opinion. I am currently running two sets of MiNewt x2 Duals, one on my handle bars and a second set mounted to my helmet. That's 600 lumens of motorist blinding power for those times when you just need to get you point across, like when they turn right in front of you despite the fact that you have a green light and the crosswalk sign
Second you really need to get your freehub body winterized. This is a overhaul of the mechanism that allows your cassette to spin free one way and engage the drive way. I have seen factory freehub bodies freeze at temperatures as high as 25 °F. That's pretty high considering it regularly gets about eighty degrees colder than that here (by regularly I mean at least once a winter). If you don't get this done, there is the potential that your freehub body won't engage and when you pedal forward, nothing will happen. Even worse, there is the higher potential that your freehub body will partially engage and break parts of the mechanisms inside your freehub body. The sad thing is that there are not a lot of mechanics around who know how to do this proficiently. Partially because manufacturers tend to discourage them from taking apart freehub bodies. I haven't ever done this myself, but from what I understand it isn't that much harder than overhauling a loose ball hub, albeit you need a special tool which Shimano no longer sells.
That brings me to hubs. Your hubs need to be overhauled with a light coating of winter weight grease, I recommend Mag-1 by Lubriplate. I use it in all the bearing compartments on my bike. You don't have to do this, because if the grease in you hubs freezes/thickens up, it won't hurt anything, but it will fight you. The same goes for your other bearing compartments. In order I would say winterize your bottom bracket, pedals, then headset. Headsets are last on my list because you may get less responsive steering at low temperatures, but you are usually riding slow enough at those temperatures that it doesn't matter as much. Pedals are an often overlooked bearing compartment for winterization, but if you get off your bike in cold temperatures and try to spin your pedals, you'll see what you are fighting against.
Somewhere between winterizing your hubs and starting in on your bottom bracket, I recommend getting a set of poagies. Lobstergloves.com has links to a few places where you can get poagies. They also have information on lobster gloves, which I don't recommend for cold weather riding. My poagies are from Dogwood Designs in Two Rivers, Alaska. They don't have a site, but are maybe the better product for it. They have evolved over the years into a fabulous product for commuting and winter trail riding, which is exactly the kind of riding I do. Many people prefer baggier poagies for expedition and endurance style racing. Bigger poagies give you more room for food and supplies. However, it is also more room to heat, room you generally don't need on a commute or two hour trail ride. As with all my recommendations, tailor your equipment to your particular riding style. For those of you out there that are fine with gloves, congratulations. My hands get cold easy and to keep them warm at -40 °F, I need mittens that make shifting and braking very difficult, if not impossible. Poagies completely eliminate this problem.
The next recommendation I have if for suspension. Get rid of it. It is just extra weight that won't work when it freezes. If you have cheap suspension (elastomer spring) it will just stop working and may work fine when it thaws, or may get extra springy when the elastomer cracks and begins to fall apart. If you have expensive suspension (oil or air sprung) there is a good chance that you will blow the seals on your suspension and permanently damage it. I don't swap my summer fork on my bike until it is consistently 35 °F out. All this being the case you can save yourself some trouble and weight by getting a rigid fork. A decent rigid fork can be had for under $100 and is well worth it considering the price of a replacing your suspension fork. I do however recommend a carbon fork if you can afford one. Besides being lighter, carbon will help reduce the vibration that can come from riding on snowpack. I won't tell you that carbon bars and seat posts are necessary for winter riding, but they do make it more comfortable.
I have a hard time placing my final recommendation for you, tires and wheelset. For first time commuters I definitely recommend studded tires. Until you get used to riding on patches of ice, they will help keep you from visiting your local emergency department. A good set of studded tires are expensive (think $100 a tire) but still less expensive than an ambulance ride. I recommend Nokians. Nokian uses the same auto grade studs that you find on car tires, so they last. If you are riding mainly trails, are a horrible wight weeny (all those little metal pieces add up), or very confident in your handling skills, then choose your favorite wide tire and experiment. I personally run WeirWolf LTs by WTB for everything. However, I am still young(ish) and spry(ish) so I expect to recover from the falls that I take every two months or so while commuting in the winter. That may get change as I get older and bust a hip for the first time. Or worse yet, if I actually fell hard enough to hurt my bike (I'd have a new set of studs the next day). As for wheels, wider rims are better. Bikes like the Surly Pugsley, the Fatback, and the 9:ZERO:7 are the current conclusion to this line of thinking. However, without needing to buy a whole new bike (which is never a bad idea when you are a committed cyclist) there are several options for wider rims that will fit a standard mountain bike. I have a set of beautifully built wheels (thanks Jeff!) laced to SnowCat rims. Several other companies also make wide rims. However, some are aimed at the downhill crowd and are way overbuilt for winter commuting or trail riding (read heavy as all get out here), so you want to pay attention to intended use when purchasing to build a wheelset. Wider rims will make a difference even with the same tires you are currently using because they will spread the tire out and give you a larger contact patch. 29'ers also give an advantage in this area. While many people don't see the need for them, they have a very legitimate place in the winter cycling world. After you have the tires you want, pressure is the next thing to worry about. If you plan to run super low pressure (I have run 3psi before) you want to make sure to glue your tire to your rim. This will prevent your tire from creeping around your rim and sheering off your valve stem. Gluing means using a contact cement type substance like Tubasti (by Velox, sorry couldn't find a website) to adhere one side of the tire to one side of the rim in a couple places. This will allow you to still change flats on the fly. Pick either drive or disk side and do it consistently to make your life easier. As far as running super low pressures this works well for flotation at higher temperatures, however at some break point (seems to be about -10°F for my Weirwolf LTs) your tires will get stiff enough that the energy required to flex the sidewall as the tire rolls will make you noticeably slower. At this point it is advisable to jump back to a higher pressure and sacrifice some traction for speed. Usually the the ice gets sticky enough at these temperatures that it hasn't been a major problem for me.