I tried wearing a raincoat to keep out the rain, but find I get hot and sweaty inside. So my question is, what clothing do you find works best for you when cycling in wet weather?
Lightweight polyester or microfiber materials do the job. Pearl Izumi is my favourite. Bike specific rain jackets are a must if you are looking for comfort. Generally they are very thin and have air vents in them. There is no reason to wear heavy rain jackets. Instead, layer up up with a base layer (often merino wool or synthetic thermals) + mid layer + the shell (which is the jacket). That is probably the most common layup for cyclists and it will keep you warm.
I bicycle year round in the Pacific Northwest and follow the mantra: there is no inappropriate weather, only inappropriate clothing. (Well, snow and ice may be inappropriate weather for bicycling) Despite commuting daily, I only purchased dedicated cycling rain gear yesterday -- and only did that for better visibility and more convenient pocket locations.
Generally: Layer so that you have options. Start out bundled up. After 5-10 minutes, you'll be warm and want to remove some clothes. When you do, outer layers should pack well. Alternately, you can just slow down. Some commuting days, I will just ride slow enough that I don't sweat. Riding a little slower only makes 10 minute difference in my commute time. I could often spend more time fine tuning my gear (and much more time parking a car).
For raincoats, something that vents well is essential and will minimize sweat. No fancy fabric will do as much for sweat reduction as a zipper under the armpit and zippers that can open the coat from the bottom. I used a multi-purpose Marmot rain coat in this configuration for a decade with very little sweat. You will also use the raincoat on cold days to cut down on wind-chill.
Visibility: If you use a general purpose rain jacket, it may not be bright. You can get reflective vests of various styles from bicycle shops. These have the advantage that you can use the vest when it is too hot to wear your high-visibility rain jacket.
Gloves are essential on days below 50F. I just used ski gloves in the winter. I got some waterproof cycling gloves that I'll try this winter.
Long underwear (as used for skiing) will extend your cycling season by months.
My legs and feet usually warm up fast. As noted by Kibbee, make sure your socks are not cotton. They will stay warm even if wet. I bring an extra pair of socks for the way home to avoid the yucky feeling of putting on wet socks. Much cheaper than waterproof booties.
You are getting some good advice here on rain jackets, but the part of the accessory for cycling in rain that I find critical are a good quality overshoes - especially for commuting. There is nothing more annoying for me that arriving to work and have the shoes and socks wet, there is no chance for them to dry completely before the journey back home so you will end up putting the wet stuff on.
I tried some cheap waterproof overshoes first but they did not really work in any heavier rain wet conditions, basically after 10 minutes you were the same as not wearing them at all. Then I bought overshoes made of neoprene, and they seemed to work a bit better, but again after about 15 minutes in the heavy rain the shoes were all wet - to be fair they kept the feet warm, so were better then nothing in the winter but still did not serve well the purpose of keeping the feet dry.
2 years ago I eventually invested in the Gore Tex overshoes and finally it is something that actually works. I have about 35-45 minutes commute journey and so far they kept me dry. So my advice would be to invest in a quality gear from start- it will save you the annoyance and in the long run the money as well.
Rainclothes designed for cycling, preferably in a material that "breathes" (Goretex being the best known, I think). You will still get wet from the inside if you go too fast, but this type of material is way better than "non-breathing" varieties.
I have a Gore Bike Wear Path jacket and pants that I am pretty happy with. Rainclothes for biking is essential in rainy Denmark :-)
Biking raingear is always a compromise. Do you want to get wet from the rain or from sweat, or a little bit of both? I use a Foxwear jacket and pants, and I stay dry if I go slowly and don't work up a sweat. The material is warm but somewhat wicking and water resistant. In the summer I'll be more likely to change into lycra for the ride and just get wet, changing when I get to my destination.
I usually just put up with the wetness. The best are clothes that don't absorb water, and therefore don't get heavy when they get really wet. If I'm going to work, I have a change of clothes anyway, and otherwise I'm usually just going out on a recreational ride, and will be back home after I'm done. One thing I would recommend is a pair of gloves with good grip. Personally, I use MEC CyclPad gloves, as they are quite cheap, and provide quite a bit of grip from the rubberized palms.
And, of course, once you've got where you're going, you'll feel a lot better about yourself if your belongings are dry.
I have an excellent Ortleib, courier style bag, which is totally waterproof, sealed by folding over the top. If you can change into dry clothes (in particular dry socks and shoes), embracing the wet is a lot easier.