What do you usually do if you spawn in a snow biome?

Usually i start running to another biome, getting some wood on the way. Then, find some sheeps for wool and make a bed: when you find a decent spot for your first base (most of the times that will not be my main base) put down the bed, lit the places with torches, make some shelter around with dirt or cobblestone and you're done for the day!
If you want to survive in the biome without moving, then i suggest a fast trip to near biomes to get some saplings, seeds and wool.

I usually make a shovel and collect it for a while and enjoy the winter calm air

I try to find a village for some food, snow biome villages usually have some cows in the pens, so that's good.

If it's a nice looking spot I'll probably try and build a home there.