H&M do some really nice lined cord trousers - I have bought DS a couple of pairs every year, and they kept him cosy even in Canadian winters (obviously with a snow suit over the top).
For tops, I layered a vest, long sleeve tshirt and jumper/hoody on the coldest days. Peeled off layers as we came inside. Toddlers seem to run warm, so DS used to just wear a vest and long sleeve top indoors.
Definitely snow boots - Baby Bogs are lovely and cosy, properly waterproof, and super cute. Often in the sales or on Amazon. I wouldn’t pay full price, they are like £50 over here.
Snow suit/fleece lined puddle suit. H&M do pretty good ones, as do John Lewis.
Waterproof mitts - again, H&M ones are cheap and work well. DS was wearing his out to play in -10C snow, and his hands stayed warm and dry.
Lined hat that covers his ears and stays on. We have had lined waterproof trapper hats, and fleece lined beanies, and both worked well.
At this age I wouldn’t bother with snow pants, just a snow/puddle suit, but for two year olds I found separates worked better. DS wore his warm coat from end of October into spring, but swapped between snow pants and lined rain pants depending on how cold it was.
And yes a liner for the push chair is essential - they get really cold really quickly when they are sitting around not moving much.