Dress Code
Dress warmer before a race and much warmer after than during it. Take more clothes than you'll need while running. I've never suffered from taking to many clothes to a race, because I could always deposit any excess. But when I did the opposite, I got very ill.
Also, make sure you dress up soon after the race, even if it means you'll sweat a little. After longer races, half-marathons and up, there are free thermal blankets to take. You can use them instead of your own clothes for a while. After a race, the body is weaker and prone to diseases. Don't risk it, dress warmer.
When running in cold weather, focus on covering up the whole body evenly and skip very warm clothes. Get a jacket, preferably a windbreaker, and some pants, but complement them with a scarf, gloves, socks and hat that covers ears.
Scarfes are awesome for running. I personally enjoy Buff and many of my friends do as well. They aren't thick, wrap well around the neck and will stop wind.
Compression clothing works really well during winter runs. It's tigher and more comfortable and increases the body's temperature after running just a few meters. After I started running with compression tights and shirt, I never went back. Many of them have extra compartments with zippers for keys and other small items.
All clothes used for running should be specifically made for sports. Otherwise, they'll soak in sweat and become smelly really fast - with a smell that'll haunt your dreams. Yes, you will sweat when running in winter.
Nutrition, Technique And Gear
Reduce water intake. You don't need to drink as much during a winter run.
Watch out for ice and tight corners. For runners, it's very easy to sprain ankles. Slow down to a walk if you need to and slowly avoid any tricky areas. When running on a lot of snow, invest in winter running shoes. They have thicker soles and sides that secure against sprains.
Running in below freezing temperatures can cause phones to shut down. Put them inside something.
Earphones allow better focus on running but also isolate ears from cold wind and can notify of progress. This avoids having to clumsily take the phone out while having gloves on and the phone being behind a zipper. Electricity-conducting gloves have never worked well for me and earphones are a decent alternative.