It is the warm up in reverse, but longer. Slow down your motions and lessen your intensity for about 10 minutes before you stop. You are cooled down when your skin is dry and cool to the touch. The best rule for exercising in cold weather is to dress in layers so that as your body temperature rises, you can take off one layer at a time. The same rule applies indoors, except that you don't need as many layers.
Proper shoes are also very important.
As sentimental as you may be about an old pair that looks worn, replace it with a new one. Shoes are designed for specific activities. For instance, a running shoe bends at the ball of the foot and has a thick heel wedge to tilt the foot forward, a shock-absorbing midsole, and an outer sole with deep patterns. The upper part of any shoe should breathe, and the whole shoe should be lightweight. Shop for shoes late in the day, when your feet are larger than in the morning.