EDIT: We've been inlining them for a while now, using comics.com's helpful embed code. Thanks guys!
Peanuts' first strip was published on October 2, 1950. It introduced the now-obscure characters Shermy and Patty, but also brought us Charlie Brown.
At the start there were only these three characters plus Snoopy, who was introduced in the third strip, apparently with a flower growing out of his head. (Now I see that it's tucked into his collar. Still seems a bit weird, though.) All the other characters were introduced later. Violet showed up early in 1951 and quickly settled in with the other characters. Later on she became mostly interchangable with Patty, but in these days she was slightly more matronly; an early sequence focused on her making mud pies for Charlie Brown.
Of the other major characters, Schroeder shows up in 1951 as a baby. Lucy also first appears as an infant, one with (for Peanuts characters) freakishly round eyes. Linus, Sally and Rerun were likewise introduced as very young children. Back in 1950-1, Shermy, Patty and Violet seem to be about kindergarten age, while Charlie Brown is too young for school. In the first year school only figures in a few strips, and only one's actually in a classroom.
Generally, Peanuts characters would age until they got to the first grades of school, then would stop. As a result, Shermy is older than Charlie Brown, who is older than Lucy, who is older than Linus, who is older than Sally, and the characters recognize this, but in the strip they are all generally the same age. There are a couple of exceptions: "Peppermint" Patty seems to be a little older than the other kids, and Rerun has to remain fairly young to distinguish him visually from Linus.
The strip was heavily stylized right from the start. In fact, the characters were arguably more stylized in the early days than they'd be later on. Look at the size of those heads! They're almost like melons on sticks. However, Schulz was far more likely to present three-quarters perspective in those early days, with much of the visual appeal of the strip coming from these distorted, yet charming, children being represented in more normal surroundings.
Sometimes this necessitated a bit of cheating: in the third panel of the second strip, Patty's punching of Shermy is done with a whir and a "whop," probably because it'd look weird to show her arm stretched out that far.
The second strip has Patty outright punching Shermy, and the fourth has her taking his umbrella! Until the advent of Lucy -minor chord-, Patty and/or Violet would usually get assigned the job of Thurberesque female spoiler.
The fifth strip is Snoopy's second. He's almost unrecognizable in these early days! Notice particularly his pointed snout. Actually, if you look carefully it actually seems to be rounded in those days, but the nose's positioning at the end makes it look like it comes to a point.
This one's particularly interesting due to the shifting size of Patty's head. From the front she looks fairly normal, but from the side it's huge. You might think this is because their bodies are wider than they are deep, but this isn't the case, in other situations their bodies are roughly cylindrical in proportion. The reason Patty looks different in the jump rope panels is her arms have to be longer in order to accomodate the rope being above her head. I do not intend to imply, by the way, that this is a weakness in Schulz's art. Any time you have greatly distorted characters you'll eventually have to cheat like this. It is an aspect of Schulz's genius that he was able to change the distortion in such a way that it's barely noticable when the need demanded it.
This strip implies that Snoopy's original owner was Shermy! Actually, although Shermy has the strongest claim to ownership of the dog, most often giving him baths and walks, the other characters do too, and he hangs out at their houses nearly as if he were an additional kid. Later on Schulz rewrote history, making him the property of Charlie Brown, who got him from the Daisy Hill Puppy Farm. (This information comes from the Metafilter thread.)
These strips come from the earliest age of the strip, when it was focused primarily on one-time gags and if there were a story it'd be a simple sequence of related jokes along a theme. Creating a comic strip is, IMO, a grueling occupation. Being funny once in a while most of us can handle, but being funny frequently is hard enough that people make their livings as stand-up comedians. But even comedians don't come up with entirely new acts every time they perform, and they don't have to draw their material either. I cannot name a single comic strip artist who kept up a daily pace for more than twenty years without falling into a Beetle Bailey-like rut except for Charles Schulz.
Some people (especially in the Metafilter thread) made the comment that the later days of the strip were a pale reflection of the classic days, and it is difficult for me to deny this. But what can be done about it? Bill Watterson chose to end Calvin and Hobbes rather than let the strip fall into the same kind of disrepair, but with the result that it is now exiting the public consciousness, in danger of being drowned out by those horrible truck stickers. On the other hand, Blondie is now a creature so divorced from its roots that they are unrecognizable; it was originally a comic serial about a gold digging flapper and her playboy paramour.
While Peanuts may not have kept up its brilliance for its entire run, really, what could? It is my opinion that even at the end it never degraded into Blondie territory, and Blondie itself is still better than many of the rest on the comics page. I am not sure if I'll be able to maintain interest in a Peanuts blog after the classic years are through, but there is a lot of material to get through before then! I intend to savor them while I can.
EDIT: Fixed a couple of links, changed a little formatting, performed some other minor edits, and inlined the strips.