According to the Mayo Clinic, a common error that athletes make while exercising in the cold is dressing too heavy. Wear layers! Your body is going to generate a lot of heat once you start moving so you will need to wear layers that you can remove to release some of that heat. Once your body cools down, you can add them back on. Make sure the first layer you put on is a synthetic material that will draw the sweat from your body. Then add on insulating material such as long sleeve cotton t-shirts and sweatshirts. Your last layer should be breathable and waterproof. Depending on how cold it is, you may need to grab a scarf or other type of neck warmer that can cover a portion of your face.
Don't forget the gloves and hat! Your fingers and ears are more sensitive to frost bite so they need to be protected. Also doubling up on socks is a good idea to keep your toes nice and warm.
I think it is great to try and get as much fresh air as you can, but there are times when it is better to stay inside. The Mayo Clinic advises to stay inside if the temperature drops below 0 degrees F or if there is an extreme windchill.
Be sure to check out the full article here - Exercise and Cold Weather- Tips to Stay Safe Outdoors.
Stay warm and safe!