May 30, 1951: Schroeder's first appearance

He doesn't have his piano yet, doesn't have his striped shirt, barely has any hair, his personality consists of  blank stare, and can't even talk, but it's the same Schroeder who would later idolise Beethoven and fend off Lucy's advances.

Oh, how much this strip changed over the years.  Most comic strips, those that were ever any good, start off great and trail off over the years, as the need to continue bringing in an income overrode any considerations of quality.  Peanuts went the other way, starting off sharp but gaggy, then gaining profundity.  Its characters changed so much since 1951, they're like different people.  (I had to stop myself from saying "between then and now," it's hard to believe that Charles Schulz has been gone for over nine years.)

May 22, 1951: Suppertime: A love affair begins

Man, that's one irate dog.  Until the magic word is spoken, of course.  This is the first time Snoopy reacts to the word "suppertime."

May 21, 1951: Those were the days

Here's another example of how times changed since Peanuts began.  We haven't had a draft in the U.S. since Vietnam.  (Mind, there was no draft going on at the time this strip saw publication, either.)

Note: Vooodooo84 notes in comments that there was a draft at that time, for the Korean War. My mistake.

May 18, 1951: Prototype of Calvin

Charlie Brown breaks the fourth wall.  To answer his question, yes, we would.

Bill Watterson has listed Peanuts as an influence on Calvin & Hobbes, and the similarity is remarkably close here.  This is exactly the kind of thing Calvin would do, right down to gloating about it to the reader.  It seems to me that early Peanuts is probably a greater influence on Calvin & Hobbes than later strips.

May 16, 1951: Beware of dog

Snoopy's a nice touch here, although is it just me or does he look like was added as an afterthought?  It has to do with his mouth overlapping the ice cream in the last panel.

One of the more pernicious influences of Peanuts is a number of cartoonists who think it's acceptable to make clip art of their characters in all their poses and just paste them into the document. Schulz was far too much of a craftsman to resort to this, but he did admit that he had a number of stock poses he'd most often draw the characters in.

Snoopy's appearance in the last panel here looks like it was just overlaid upon the frame, but I think it's more likely that Charles Schulz just lightly inked the overlaying portion of the cone. If the cone were drawn to the point where Snoopy's snout directly overlaid it it'd be harder to read. You can occasionally see the light inking idea in other strips in the early era, when a character is standing in front of another, for example.

May 15, 1951: Oh he's a clown, that...

Charlie Brown's looking mighty expressive here.  Later on, he'd be more likely to just suffer with a sigh instead of complaining about it.

CB's name has been, by far, the most mentioned among the strip's meager cast so far, which is weird because it's the longest.  It's interesting that now, over 50 years after this strip's original publication, you could use "Charlie Brown" as an insult and people would generally know what you meant.

May 14, 1951: No more mud

Ah! Could this be the end of the mud pie jokes? (Hint: no.)

May 5, 1951: Groan

Another chase.  I have to disagree with Charlie Brown in the last panel, that was not really all that notable a joke.

May 3, 1951: Before "sugar rush" entered the vocabulary

The Snoopy-hears-food pattern is formula; the varition is that Charlie Brown's already eaten all the candy.  Snoopy would even do this much later.

Note Charlie Brown's round eyes in frame 3.  Lucy starts out with those as standard equipment but they don't last long.  The parenthesis around her and Linus' eyes (I call them "Binkley eyes," after the Bloom County/Outland/Opus character) are a remaining vestige of that.

April 30, 1951: Pennies went farther then

Patty wears a travelling coat in this one, a decidedly 50s-ish touch.

This is the start of a minor running gag, the Peanuts kids often get candy or comics from this particular drug store, and the manager gets annoyed when they don't buy things. (Just think: Peanuts started back in the days when drug stores were still a suitable kid hang-out spot, and ended in the days of the Internet!)