There are a ton of online articles offering advice on how to stay comfortable when running, walking, or hiking in cold weather conditions. I am going to offer some insider tips from my 30+ years of exercising outside.
Warm Up Your Gear
If you keep your shoes/hats/gloves/jacket in an area where the temps drop make sure you warm them inside before you put them on. If you put on a pair of shoes that have spent the night outside or in the trunk of your car in cold conditions they will never warm up during your run. If it's really cold out warm your gear by a vent, fireplace, or heater before they go on even if they have been stored indoors.
Kinesio Tape; Not Just for Skin
Modern running shoes are built with lightweight mesh that is very breathable. It's great for hot, summer days but not so great when the temps fall. To cut down on the air flow over your toes cut a piece of kinesio tape and apply it to the top of the outside toebox of your shoes. This will eliminate a lot of the cold air that enters around your toes. If the tape isn't subjected to a lot of moisture it will last for several weeks of running.
Cover Your Neck
A neck gaiter is almost like putting on another layer when it's cold. It cuts down on the chimney effect of warm air escaping through the neck of your shirt or jacket. If you get too warm it's easy to remove and tuck into a pocket or the waistband of your tights. I use a very thin gaiter most of the time, saving the fleece model for very cold temps.
Protect Your Face
Your face takes a beating in cold conditions. I recommend applying a very thin layer of petroleum jelly over your face and lips prior to heading out.
Warm Up Inside
Many people find it helpful to do a short warm-up before beginning their run or walk. This could be some light, dynamic stretching or easy walking to get the muscles loose. In warm weather it's great to do these outside but in winter I do them inside where it's warm. When I walk out the door I am ready to get moving and generate some body heat.
Tuck in Your Baselayer
It sounds simple but tucking your baselayer into your bottoms helps seal warmth against your skin. Bonus: If you start to overheat just untuck and let some of the warm air escape.
Various warming creams and gels can make cold weather a little more tolerable. Items that contain menthol or capsaicin can have a warming effect on your legs, lower back, and neck.
Dress for the Effort
Harder efforts will require less clothing, easier efforts more clothing. A long, slow run with walk breaks is a totally different activity than a hard 4 miler. Make sure you figure effort in when choosing your gear.
Start Out Cold
This has been said a million times but it begs repeating. Start your run a little cold and after about 10 minutes you will be fine. I see a lot of people running with a jacket tied around their waist during the winter months.