The great downside of living in New England is the weather. Six months of the year, you know it's going to be cold and wet, and the other six, you can reasonably expect that it will be cold and/or wet. A good friend asked me for some information on riding through the winter, so I'm replying to him here.
In short, the challenges of winter riding are: staying warm, staying dry, and staying clean. Doing all of this on a budget is an additional challenge. I'll propose some handy tricks for doing so here.
In short, my tips are: get good, warm clothing, wear Gore-Tex or similar, install fenders, and wear shoe covers.
Staying Warm (Managing Heat)
This is the most obvious challenge of riding in cold weather. You need to stay warm over a range of temperature differentials. You need to not freeze when you first head out, but not overheat when you are working hard.
The human body loses the majority of its heat through the head, as powering the brain uses a substantial portion of your blood supply. As a result, you should certainly keep your head warm. Muscles will be more apt to keep themselves warm, since their surface area to volume ratio is fairly low, and they increase in temperature during operation. Correspondingly, your joints, genitals, and extremities will get the coldest, since their surface area to volume ratio is relatively high, and there is less tissue there to absorb heat.
Cyclists will always need to wear more clothing than runners in cold weather, due to the effect of wind chill. A trained cyclist going 18 MPH is working about as hard as a trained runner at roughly 7:30 pace, or roughly 8 MPH (though I may be mistaken?). That difference alone is responsible for a roughly 5 F difference in perceived temperature.
My rules are: keep the knees and elbows covered below 68 F, wear long gloves and a hat below 60 F, wear warmer socks and tights at about 50-55 F, start adding more layers below 45 F. Below 30-35 F, I want to make sure that every square millimeter of skin is covered with at least one layer.
Preferred clothing in cold conditions is made of either merino wool or synthetic materials, is easily removable to facilitate dressing in layers, and in the case of a garment for the upper body, has a long zipper to allow temperature regulation. Gore-Tex, a trademark for a type of synthetic microfiber fabric, allows gaseous but not liquid water to pass, which in turn, will allow your body to breathe while keeping you warm.
Recall that clothing works to keep you warm by insulating you through keeping air close to your body, and by preventing heat from escaping by being constructed to keep that warm air from escaping. A counterintuitive result is that, when you enter a warm room after being out in the cold for a while, you should undo as much clothing as possible. This will allow warm air to come in contact with your skin, and allow your clothing to "recharge". Just think about how warm a fluffy towel is when you take it out of the dryer, and how long it can stay that warm.
If you have one of those water bottles that advertises keeping beverages cold in hot conditions, realize that it will also work to keep beverages warm in cold conditions. Bringing hot tea on a ride will help!
Staying Dry
Water is your enemy in cold weather. Be it from internal sources (i.e. perspiration) or external sources (i.e. road water, rain water), water will only make you cold and miserable. Going back to the theory that warm air in the voids of your clothing keeps you warm, we can easily see why. Water will fill all of the air voids in your fabric, replacing warm air with cold water. Since it has such a high heat capacity, and a correspondingly high heat of vaporization, cold water will suck all of the warmth out of your body in an attempt to equalize the temperature between the water and your body.
We want to be able to keep dry by repelling water, while allowing sweat to evaporate. Gore-Tex fabric accomplishes just this, as mentioned earlier, though there are plenty of non-trademarked fabrics that achieve this goal as well. Having a breathable, water- and wind-repellent outer layer is the best you can do. Note that most low-end rain jackets do not breathe, and will make you get wet from the inside!
It is important to supplement your good outer layer with a quality base layer. Synthetic fabrics and merino wool will retain much of their heat capacity when wet, whereas cotton will not. Therefore, if it is very cold outside, and particularly if it is wet, do not wear cotton, at all!
If the roads are wet, your feet will very likely soak through. To prevent this, you can get shoe covers or "booties" to prevent water from entering your shoes. If your feet do get wet, change socks as soon as you can, and work to warm and dry your shoes using old crumpled newspapers to transfer most of moisture from the shoes.
Staying Clean
If you are riding in the winter for the purpose of transportation, staying presentable is a priority. Most of the tips I have recommended above will keep your base layers fairly clean, but to be sure, there are additional steps you can take. These tips are most helpful if you cannot shower or change after a ride and before your various non-cycling obligations.
In addition to wearing shoe covers, consider wearing an additional pair of leggings over those you wish to keep clean. Your ankles and shins can get muddy and wet as well.
In the winter, when snow and muck are everywhere, a rear fender is a must. The most rudimentary variants of fenders attach to your seat post, and prevent your rear wheel from spraying muck all over your backside. Fenders that more closely follow the contour of your wheel, and either attach to your seat stays and fork (for racing bikes) or to special mounts (for bikes with enough tire clearance) will do a better job of preventing road spray. A mud flap for the front fender will do even more to help keeping your feet clean.
Conclusion
There is plenty you can do to keep riding through the winter. Good clothing and ample preparation are your friends. Consider warm clothing to be an investment, not an expenditure.
There is no such thing as excessive cold, only insufficient clothing.