Bookhunter's opening chapter is unrelated to the larger plot, and instead serves to set the tone of the story to follow. A SWAT team has been called in to deal with a freelance censor, and when they invade his apartment, they find the perp threatening to immolate himself and the books he stole; clearly, in this universe, books are Serious Business. Detective Bay's response to the standoff shows that, underneath his veneer of rationality, he is a Cowboy Cop willing to do completely insane things in his pursuit of criminals.
The first dozen or so pages of Runaways exist solely to provide one of these for each of the kids: Gert is shown correcting her dad's Latin in the middle of an argument about politics, Chase is shown getting beaten up by his dad for getting bad grades, etc. Similarly, the first time we see Ascended Fanboy Victor, he's talking about superheroes with his buddy Jorge.
Issue #1 of The Fantastic Four: It all begins with one strange man shooting a signal flare into the air — a flare announcing, "The Fantastic Four!" Sue Storm was hanging out with a socialite friend at the moment the flare went off. She promptly turns invisible and walks out into public, testing her powers by taking a cab and attempting to pay the unsuspecting cabbie the fare. Ben Grimm, who is at a clothing shop in heavy disguise, hears of the signal flare and sheds his constricting clothing, revealing himself as the monstrous Thing. His debut catches the attention of the police, who open fire on him and thus force him to escape through the sewers. Johnny Storm is at a service station getting his car fixed when he learns of the flare (which by now had turned into the number 4). He flames on and flies through the air as the Human Torch. The Nation Guard attacks him with a nuclear warhead hunter missile. Two stretchy arms grab the missile before it could obliterate him and disposes of it via the sea. By now Johnny's flames had worn out and is plummeting to his death, but the strange man from before saves him, revealing himself to be Dr. Reed Richards, AKA Mr. Fantastic. And this is all before the Superhero Origin!
Amazing Fantasy #15 is all about this. It starts with a shot of Peter Parker being snubbed by his friends, then Uncle Ben playfully teasing his nephew and Aunt May feeding him a healthy breakfast. Then Flash pushes Peter over and steals the girl he was introducing himself to.
In the first issue of the series, J. Jonah Jameson comes onto the scene writing one of his infamous anti-Spider-Man editorials. It could be said, however, that the real Establishing Character Moment is the day after Spidey successfully saves Jameson's son from a malfunctioning aircraft, when he runs an article about how Spider-Man sabotaged the aircraft himself and broke several laws in rescuing his son.
Subverted with Gwen Stacy; she shows up flirting with Flash Thompson and vaguely intrigued by Peter Parker, but after some Character Development, the scene actually seems out of character.