The activities of a 'Snow day' really depend on the age group you focus on. Younger kids tend to play outside in the snow, sit and watch TV, or play video games. While high school level students are usually with a group of friends. Sometimes high school level students get together and go to a movie later on in the day (depending on weather), or go out to eat.
But mostly, at least around where I am in Southern Maine, it's just a day to relax. A lot of people regardless of age sleep in relatively late, and then just find something spur of the moment to do. Unless there has been a lot of Snow for a couple days, you usually find out the morning school is canceled, so there isn't a lot of time to plan.