Big Lipped Alligator Moment: Tons. It's kind of justified, considering that it was the first full-length Peanuts movie, but it sort of goes overboard on it. The skiing scene with Snoopy in New York City and Shcoreder playing that Beethoven piece are major ones.
Ear Worm: "Champion Charlie Brown". You'll find yourself humming it a lot, partly because it's sung when Charlie's at his high point. The song even makes a cameo during the opening Cinema Center Films logo.
Failure is a part of life and it's unavoidable at times.
Another Aesop is "Time heals all wounds (no matter how badly you mess up.)"
Linus: Well, I can understand how you feel. You worked hard, studying for the spelling bee, and I suppose you feel you let everyone down, and you made a fool out of yourself and everything. But did you notice something, Charlie Brown?
Charlie Brown: What's that?
Linus: The world didn't come to an end.
Nightmare Fuel: The Beethoven sequence. Sure, it might be lovely to watch for someone who appreciates the artistic value, but otherwise....
Society Marches On: Charlie Brown twice says to himself, "It's Sydney or the bush," which in the U.S. was a very short-lived idiom. It makes more sense in Australia.