Dark Avengers: Dark Avengers: Ares introduced the titular character while he subjects a group of soldiers to Training from Hell by getting drunk while shooting at them with a Gatling gun.
Fantastic Four: Fantastic Four #1 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!
New Mutants: New Mutants: Lethal Legion reintroduces an old Marvel villain to a new audience. Count Lucchino Nefaria first appears in his mansion, standing in front of a huge portrait of himself and wearing an old-fashioned suit, a waistcoat, a cape, and a monocle. He plays a vinyl record (implied to be opera or classical, given the image on its sleeve) and fills his wineglass from a decanter. He laments the fact that people have forgotten to be afraid of him these days. And then he brutally executes an intruder who's tied to a chair. All of which establishes that he's an old-fashioned card-carrying villain... but a genuinely dangerous one.
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.
Amazing Fantasy #15 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 Amazing Spider-Man (1963) #1, 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.
Spider-Gwen on the cover to Edge of Spider-Verse #2 has her changing into her costume, setting the comic book world ablaze.
Mary Jane Watson's "Face it Tiger, you just hit the jackpot!" line, which she utters in her first appearance, makes quickly clear that she is a cheerful, feisty, bold girl.
In his very first appearance, Carnage kills a man by suffocating him and forcing tendrils down his throat, while explaining to his victim that he chose him by literally flipping through a phone book until he found somebody he thought had a stupid name.
Magneto is introduced when he fearlessly attacks a horde of killer robots. He goes on to lament the deterioration of his power over metal and simultaneously establish that he is still a formidable opponent: these facts set up his character and plot arcs.
Cyclops interrupting a scene of Police Brutality and insisting that the victim has human rights.
In the original series, Beast (Marvel Comics) is an odd delayed example. The archetypal Genius Bruiser was introduced smoking a Distinguished Gentleman's Pipe and reading advanced calculus in his first appearance... of the third issue. In the first two issues he is a hot-headed teenager, very different from the characterization that would make him famous.
X-23: Laura - despite her young age — has a lucid nightmare involving blood and corpses. She calmly explains the details of this dream, ending with:
