Our family has never lived in a place where it’s supposed to snow (we saw some while in Dublin but it was very unusual) but we usually visit snowy places at least once a year. I did decide a few years ago to invest in some proper snow pants for the kids when I found them super cheap but before that we had to improvise for snow play.
Layering the warmest clothes is always best. A pair or two of socks over the warmest shoes. Pajama pants under warm pants. T-shirt under sweatshirt/sweater under coat. Two pairs of gloves, especially if not waterproof.
And here are a few ways to stay dry when you don’t have proper waterproof snow gear:
Plastic baggies and rubber bands. Put on your child’s warmest socks and shoes and then plastic baggies over, secured with rubberbands at the ankles to keep everything dry (though be careful with walking, the baggies are very slippery on ice!)
Fleece. If you don’t have proper snowpants, fleece pajamas are a great alternative as the fleece is extremely warm and repels water. I usually layer the fleece over long-pajama/waffle fabric pajamas for extra warmth. Jeans are a close second if you don’t have fleece for water-proofedness/warmth in the snow.
Raincoats. If your child has a warm coat that isn’t waterproof, simply layer an adult raincoat over the top for wet protection.
Ski gloves. If you have really little ones, adult ski gloves are actually perfect as a boot replacement, slip over slippers or a few pairs of socks for fun toesie boots!