September 27, 1951: Cigar-Box Banjo

Schroeder actually doesn't do much in the past three strips other than play the piano. The humor comes mostly from the other characters reacting to it. A tremendously silly thing is not as funny as people dealing with it.

As the strip gets older, the characters eventually kind of become creatures of their settings. Schroeder eventually becomes seen in few places other than before his piano, which also means that Lucy, the other resident of their little two-kid pocket universe, becomes his main contact with the rest of the Peanuts world.

September 26, 1951: Yeah, That'd Probably Be Asking Too Much

Another in the Schroeder-as-musical-prodigy series. There is also a first in this strip: it's the first time a character is represented as participating in a real-world organization or event, long before Snoopy's games at Wimbledon.

Let's note the progression of the joke:
Strip 1: Charlie Brown introduces Schroeder to the Piano. The gag: he takes to it immediately, and brilliantly. The punch comes from the suddenness of the ludicrous situation.

Strip 2: Strip beings with the ludicrous situation, set up by the past strip. The gag comes from examining its consequences. Punch is added by making it even more ludicrous, by taking the already-amazing event of a baby playing piano extremely well and making him a composer, one who's even titled his work despite being barely verbal.

Strip 3: Begins again with the ludicrous situation, but now takes it for granted. The punch comes from putting a lampshade on it; Schroeder is talented enough to tackle Braham's First Concerto but not the second because he's "only a baby," even though no baby (except maybe Mozart) could do any of these things. This also subtly normalizes the situation.

In tomorrow's strip it'll be completely normalized, and the humor will come from another character interacting with the bizarre sitation.

September 25, 1951: More Schroeder at the Keys

Look at the piano score in the first panel. Intense. Schulz had been quoted as saying he took time to make the scores accurate and loved how they looked on the page. Keep in mind, at this point Schroeder has only said a couple of words, yet he's already titling his compositions.

Another subtle innovation in these strips is a recurring gag template for Peanuts, the blatantly bizarre thing that is somehow real that the other characters can react to. Snoopy is a particular focus for these kinds of shenanigans later (I remember the "whirlidog" sequence, coincidentally also featuring Schroeder), but Lucy gets into some of it too. It works best if the bizarre character possesses an elemental, "force of nature" style of personality for it's a good way to illustrate their strength of will, by presenting it as triumphing over reality itself.

September 24, 1951: The Ceremonial Changing of the Archetypes

AH, it's Schroeder's first time in front of a toy piano, cementing his long-term role in the strip. We even get a good blush out of Charlie Brown while we're at it.

For purpose of winning trivia quizzes later, note: it is Charlie Brown who introduces Schroeder to the piano.

September 22, 1951: Why is everybody always picking on me?

Stuff like this can't be good for a little round-headed kid's self-esteem. His long-term personality is settling into place. By the way, that's a particularly Calvin-like drawing of him in the second panel.

September 21, 1951: Camping In

How big is that tent? In the first panel Patty is obviously taller than it; in the last, it's taller than her.

September 20, 1951: War of the Sexes, Tricycle Front

Wow. First the girl takes his bicycle, then she basically mugs him for his tricycle! It's almost like they're married! zing!

But do you feel it? That tremor in the ground? That chill in the air? Yes, Patty here is a rehearsal for one of Charles Schulz's great creations: the magnificent, malevolent LUCY.

September 19, 1951: Kitty

Schroeder's first word is "Kitty."

NOT "BEETHOVEN."

September 17, 1951: More forthright than Bill O'Reilly

The key here is Charlie Brown's awareness of his own mental state, and the humor derives from the fact that he knows it and yet refuses to do anything about it.

September 15, 1951: Because Charlie Brown stands in the way?

It's funny, but also seems to suggest that either Charlie Brown is incredibly accident prone, or the other characters are resigned to him standing between them and the post.