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A good pair of snow boots are a must. They need to be waterproof and have plenty of insulation to keep your little one's feet warm and dry. I'm a big fan of lightweight insulation, especially for kids. My kiddo sports Kamik's.
Here is another snow trekking necessity. It too must be water-resistant and have plenty of insulation for playing in the snow. I'm a big fan of the grow-with-your-child products. That way you can at least get two seasons use before your child outgrows it. I like Columbia snow suits.
Your child needs a warm pair of mittens. They need to be waterproof or water-resistant and preferably not cotton. Cotton traps moisture and moisture + cold = frostbite. Opt for a completely synthetic fabric, wool, leather, or fur. Here too, I like Columbia.
Always dress your little one in layers. Thermals are great. Preferably use the moisture-wicking kind. That way if your child gets hot you can always remove one layer. Even with socks you need to layer or use non-cotton, moisture-wicking material to prevent sweating. Sweating leads to moisture accumulation which can also lead to frostbite.
Be sure to outfit your kiddo with a warm hat. I like fleece hats with ear flaps. If its really cold I will top it off with a fur(or faux fur) cap. Mostly though we just throw the snow suit's hood up and roll.
Don't forget that you're out to have fun! Pack snowshoes, hiking sticks, skis, and whatever else you enjoy! We like to carry a sled so we can pull our son when he gets tired. And don't forget the Hot Hands for a quick warm-up. And, as always, you need to pack a first aid kit and an emergency blanket.
Afterwards, take the kids in for cocoa and warm up by the fire!