Check the winter forum for tips on winter camping. By winter, I mean cold weather with snow on the ground. I used to live in Northern Ohio. Winter there just meant miserable cold, wet weather, sometimes really cold as I recall, but not enough snow to be enjoyable most of the time.

For snow camping, get a copy of Allen & Mike's Really Cool Backcountry Ski Book. It is about half skiing and half camping. Well worth the price-about $10 or so online, even if you are not skiing.

For Ohio weather, your 3 season gear will probably work fine-meaning tent and stove. Warm clothes are essential, learn how to layer up and get warm, dry footwear. Sorel boots can be found at bargain prices on eBay. They are a good choice for an inexpensive winter boot. Not sure if they come in kid's sizes.

Wet and cold are your enemy. You probably learned that in the class you took. You don't need to spend a fortune on clothes or gear-the really top flight and light gear, yes, but things like wool sweaters can be found at secondhand shops for next to nothing and they work really well.

You can be ultralight in winter if you know what you are doing, are willing to compromise comfort and your margin of error for safety and spend a fortune on gear. Otherwise, going UL in winter, regardless of what you may read here, is foolish.
I drag (literally, because I tow a sled) way more stuff than most people carry because I go alone and like to be comfortable. If I was a perfect weather predictor, I could leave my tent at home. I'm not and neither is the NWS. Good, but not perfect. My winter tent should withstand most major storms in the places I go, so in spite of it being heavy and bulky, along it comes.

btw, I'm not wearing that parka in my picture just because I like the color.