We have prope snow boots from ECCO (excellent - grippy, fleece lined, supportive, goretex), but they are expensive £50/£60. Might be worth seeing if you can get a cheaper/older version on EBay (we did last year for about £35). Not worth it for everyone, but we are off skiing for 11 days.
Welles with thick socks don't cut it if you want to be out in the snow for more than 10 minutes. It's annoying, because depending on where you live, they won't get much use before your DS outgrows them but if you want to have fun in the snow, you want warm feet!