Here are some tips for winter running and clothing
Clothing and gear - head to toe recommendations
In general, it's important to have layers available in winter, and to be able to remove or replace layers as you warm up or the wind shifts. Also, you should dress such that you are slightly cold for the first 10-15 minutes of the run; otherwise you will likely be overdressed.
If you will be driving somewhere to run and will be outside for more than a minute or two after the run, it's essential to have extra warm and dry layers to use after the run. Otherwise you could risk hypothermia.
Shoes: your regular running shoes will usually be fine. If you plan to do a lot of trail running, trail running shoes can be more helpful in the winter than other seasons because they have more grip and durability, and some trail shoes have water resistant materials. If we get ice or snow, traction devices can help you run in slippery conditions. Yaktrax or something similar are good options. In DC there are usually only a few days a year with conditions that require traction. If you spend any weekends in the mountains you might encounter these conditions more frequently.
Legs: tights or running pants. Your temperature threshold will vary. I usually switch to tights in the upper 30s. Old style sweatpants (ie loose cotton) are not ideal but will do the trick if they are your only option. Designs with a small pocket for keys are a good option. Some have larger pockets that could work for carrying a phone, or for snacks if you go on a long run (you probably don't need refueling snacks if you stick with 12 miles or less). Compression socks can also add some warmth.
Torso: moisture-wicking shirt with a quarter or half zip. Cotton is not recommended for any running, but especially in cold weather since it does not dry quickly. In colder, windy, or rainy conditions, a light running jacket or windbreaker is a good idea as well. You can always take it off and tie it around your waist. For mild or warmer conditions I don't usually recommend a rain jacket (you will get sweaty and just as wet from the extra warmth of the jacket so it's counter-productive) but with rain and temps in the 30s or 40s a jacket is a wise choice. Arm warmers paired with short sleeves are more commonly used in racing than for training runs, but could be an option.
Hands: Some running shirts have a thumb hole, which allow you to tuck your hands inside. This works well in moderately cold conditions, since your hands might warm up enough that you want to take them out after a few miles. In colder conditions I recommend gloves. Simple work gloves are OK; gloves designed for running will work better. I like these from Sporthill (which also produces other good gear specific for runners). If you don't have gloves, an extra pair of socks (worn on the hands) is a good alternative.
Head: a moisture-wicking headband designed for running is a great option (these are different from the type worn to keep sweat out of your eyes in the summer). In colder conditions a full beanie-type hat works well. You can make due with a generic winter hat (like the ones they give out at Nike regionals) but the hats designed for running (like this one) will work better - more streamlined and less moisture retention.
Other winter running tips:
Visibility (as in, drivers being able to see you) is more important this time of year since we are often running near dusk. If there is snow or ice you sometimes need to venture into the road to get around bad patches on the sidewalk as well. A reflective jacket, shirt, or headband is a good idea. If closer to dusk or dawn, blinking safety lights are a good addition to your wardrobe.
Locations: In the immediate DC area (especially in recent years) we get rain more often than snow in the winter. With rain, your favorite trails might be muddy but should still be passable. With snow or ice, remember that trails will take longer to be clear than sidewalks or roads. Some multi-use trails (paved trails like the W&OD) get plowed, others do not. You might need to resort to running small loops on neighborhood streets in order to find a safe place to run. Ice and snow cleared on sidewalks will be variable too, since it's often the responsibility of the homeowner. It's important to pay careful attention to the conditions underfoot so you can avoid slipping. If running in a group, guys in front should call back to warn about hazards. (note the above suggestion about traction for your shoes).
Wind - if running in an open area with nothing to block the wind on a cold day, run into the wind first and then it will be at your back when you turn around. Otherwise you will get warm with the wind at your back and then be chilled when you turn around.
Snow - if we get enough snow that people are out skiing on the multi-use trails, please do not run in the ski tracks. This ruins the tracks for the next skiers who want to use the trail. The wide multi-user trails (like the towpath, Capitol Crescent, W&OD, etc) have plenty of space for you to run parallel to any ski tracks.