Running in cold weather can be a tricky thing because, first of all, “cold” is a relative term so dressing for “cold weather” kind of depends on the person’s perception of what’s truly cold, what’s more cool than cold, and then those people who just seem to be chilly all the time, even when they’re running outside during warmer months. The key to dressing for cold weather is layers, layers, layers. Especially if you're new to running, you might not know how your individual body responds to cold weather. But if it’s really cold, the last thing you want is to underdress and risk getting sick. If you wear layers, you can strip them off as needed. Start out by picking a running route that you’re familiar with so that you can take off that heavy vest and slip it behind a bush, or tie your jacket around your waist.
You also want to be sure to protect your extremities. Your hands, feet, ears, and toes are more inclined to frostbite because they’re usually harder to cover, and don’t have great blood flow. They’re likely to get colder than your limbs as well. So you might find that you can actually go out for a run when it’s fairly chilly and be comfortable in a shorts and short-sleeved t-shirt, but still need gloves, thick socks, and a headband to protect your ears.
Running in the cold also increases your risk for running into some nasty inclement weather. Always watch the weather and know if there is any chance of snow, sleet, or ice in the forecast. If the forecast is calling for some bad precipitation and you have ready access to a treadmill instead, although it might not be ideal, opt for the treadmill - it might save you from getting injured. If you DO risk it, be extremely cautious of patches of black ice that’re harder to see and increase your risk of slipping and falling. Bring your phone or some means of communication so that you can always call for help. Better yet - run with a partner!