I've learned a few things while commuting by bike in Corvallis, Oregon: Biking in the cold and rain can be quite enjoyable, thanks to nylon and polyester. With three layers, I'm more comfortable with temperatures in the high 30s / low 40s than I am during the summer.Base layer: I love my Patagonia "baselayer" -- their term for long underwear; I use their Capilene 3 long-sleeve shirt and "pants." They're warm and thin, very comfortable clothing while sweating, and they help me cool down very quickly. Air cuts through them easily. As the term baselayer implies, you'll probably want to put something over these; I do.
Middle layer: Any sweatshirt will do. This layer is what gets dirtiest, as your sweat goes into it, and some water will get to it.
Jacket: I use a simple, non-breathable nylon bicycle jacket, made by J&G, an Oregon company. Costing about $40, it's good, but not perfect. It's great at stopping the wind, which is very important, and you will stay dry, for the most part (which I will explain). The fleece around the neck is a nice touch, and the cut of the jacket (with extra-long arms and back) works well for biking. If I had to choose again, I'd probably go for a more-upscale jacket, which runs about $135.
There are two downsides to the cheaper jacket: 1) There is no fabric covering the zipper, and your shirt will get wet (under the zipper) if it's raining at a good pace, and 2) You will get hot during longer / harder rides, as water vapor can't easily escape; it's probably not much fun when it's warm out, like in the 70s. The underarm zippers, which I thought were a gimmick while I was shopping, do keep me significantly more comfortable when they are open -- and water doesn't usually find its way in there, so I leave them unzipped.
Pants: Polyester / nylon "exercise" pants work great when it's not raining. Combined with the baselayer above, I stay warm, but never hot. When it's raining, I use rain pants. Again, I went with a cheaper, non-breathable pair, but I would recommend going with a more-expensive, breathable pant -- $80 at the local cycle shop. Every pair of rain pants I've seen is designed to easily put on while you're wearing shoes, which is useful; they also roll up small, so I keep them in my panniers (bags).
Hat: Every piece of clothing is important, but you'll never find me on my bike in the winter without a winter hat. It keeps my ears warm and my hair dry. My hat is acrylic; I suppose any will work.
Gloves: My gloves are waterproof, windproof and breathable. About $25. I'm not sure what the "waterproof" is supposed to mean, exactly. I think it means there is some waterproof fabric somewhere on the glove, but your hands will get wet with time; however, they do stay warm. The fleece on the top is nice for wiping your nose; it may be gross, but it is nice and necessary. Like most gloves, they take a while to dry out.
Shoes: I use regular sneakers. This is a bad idea if you're not comfortable with your feet slipping from time to time. They will get beat up from the rain and general dirt that gets kicked up from the road, but they work fine. As long as you avoid storming through large puddles, your feet will stay reasonably dry; if you do decide to go through puddles, your feet will be soaked.
I've shopped for shoe covers / something for my feet, but everything I've found looks incredibly lame / excessive / expensive for short trips around town. Before Burley only made "the world's finest child trailers," they apparently made some good rain gear, including shoe covers, but they don't make those anymore.
I don't know it all, but like I said before, I think every piece is important. Expect to spend a bit of money -- probably more than you'd prefer (you can easily hit $350 before the day is done) -- but you might be surprised at how fun it is to stay out in the rain and the cold with the proper clothing.
