There is no plot or ongoing story in this Peanuts special. It seems more like an animated version of several of Charles Schultz's (who gives us an introduction to the show) Peanut's comic strips. At least there is some variety at the very least.
Some of the stories include Peppermint Patty being duped into graduating from Dog Obedience School believing it to be a Private School, the kids fussing over undone homework, Lucy throws away Shroder's piano and Chuck ends up in hospital. It's mildly amusing stuff but the changing stories will probably make it end being suited more for repeat viewings.