This is the first time we've tried painting on snow, and it turns out it's pretty entertaining. This, for us, was definitely process-focused and not product-focused. There's something really cool about watching how the snow sucks up the paint, or how the paint sinks into the snow--however you want to look at it. It reminds me a lot of getting a sno cone at a fair, when they pour bright artificially colored sugar water over the ice shavings.
I'm not sure what other people usually do, but I just made some jars of very watered-down liquid watercolors, and we brought out paintbrushes. The regular, wide, round paintbrushes worked really well. The sponge brush did not. Our snow is perfect for this right now--it has a dense, hard layer on top, so you really can pretty much just paint on it. If you had thick piles of very fluffy snow, I think you'd have to take a different approach.
Do you see what I see in the following photo? Bojey, happily engaged in a winter activity! It was actually pretty warm (relatively warm, I should say), so her mitten refusal didn't affect her too much.
After Bojey went down for nap, it was snow painting, round two, complete with new full jars of paints, for Ninna and our two neighbors:
I recommend trying this if you've got the right kind of snow. I was a little skeptical, but now that we've done it I'd definitely do it again.