I chose these two sketches to start with because they give a really good impression of who the characters are if you don't happen to be readily familiar with them. The first video shows the lengths to which Ernie will go for a joke, and the second video shows how enthusiastic Bert gets over his "unusual" (i.e., boring) hobbies. I guess collecting bottlecaps isn't necessarily that esoteric, but he also collects paper clips.
These videos also demonstrate how widely Sesame Street sketches in general can range of the spectrum from the purely entertaining to the didactic. There's a very clearly articulated lesson to the second sketch, i.e., everyone has different interests depending on their point of view. (Also, be sure your hands are clean before touching Bert's bottlecaps.) Conversely, I can't detect any sort of lesson in the first sketch, and I think the point of it is just to be funny, although I suppose it could fall into the category of "critical thinking".
I also picked these videos because they are from very early on in Sesame Street's run; the second season, to be precise. While the characters' personalities were well established at this point, the characters' design was still in flux. The most obvious difference in these sketches versus later sketches is that, at this point, Bert's eyebrow was immobile. Around the third season or so, the puppet was modified so that Bert's eyebrow could move up and down, giving him a much wider range of expression and therefore funnier reactions to Ernie's shenanigans. Interestingly, if I'm interpreting Jim Henson's early sketches of Bert and Ernie correctly, the idea of a mechanism to move Bert's eyebrow was part of the original concept, so I wonder why it took so long for them to build the puppet that way.
Eventually, Bert's eyebrow would almost become a character in its own right. I remember being oddly fascinated by it as a child. I also specifically remember watching the first video as a child and not noticing that the characters were off-model. On one hand, it's kind of understandable since so much of the sketch is focused on Ernie's fingers, but on the other hand, it really should have been apparent with Bert's take to the camera at the end. It kind of makes me wonder how many and what changes to the characters' looks they can make before they become unrecognizable. It's kind of a scary thought, really.