I spent 16 years working with kids in Scouting and took many a class on how to stay safe in weather conditions. I really took note of how to dress in extreme cold, as I am naturally a chilly person.
First thing is layers! I recommend 3 layers.
1. Base Layer: This is best described as the Under Armor layer. A moisture wicking layer that keeps your second layer dry. This is a good investment if you workout, ride a bike or hike. Think long underwear, but not cotton.
2. Insulating Layer: This is a fleece or hoodie that can keep you warm. many times this is wool. BUT it must be kept dry.
3. Wind & Water Shell Layer: This layer will break the wind and keep the water from getting inside to the insulating layer. once again, if you workout or hike you probably already have this layer.
For me, a cold day has the base layer on from head to toe, the insulating layer from my ankles to my neck and the rain/wind shell from top of my head to my toes. My head and feet perspire a lot so I don't want my body to get wet.
NUMBER TWO! Limit your exposure! In the type of cold we're looking at for about 36 hours, your exposed skin will be damaged in about 10 minutes!
Scouting.org has some advice, too.
CLICK HERE for more info!