In Swinging Pendulum Kaien is established as a male Genki Girl who wants to be Ichigo's older brother figure in his little note to Ichigo, remainding him of their lunch date:
Morning, cousin! Join me for lunch today, okay? I’ll swing by at noon to pick you up. Your favourite big bro, Kaien^_^’
Asuka is established as a Shrinking Violet when she refuses to look Ichigo in the eye and addresses him very formally when she tells Ichigo that he has been assigned to tutor her for extra credit by their teacher, Koyonagi.
Despite being off-duty, Adam Jensen takes down a ten-Asari bank robbery single-handedly through stealth, offers himself to the last one in exchange for her hostage, then defeats her non-lethally but painfully when she still refuses to see the light. He's a stealthy One-Man Army, but he's also always looking out for others; that said, Good Is Not Soft!
The first onscreen appearance of Johann Liebert Ramsus has him concoct a brilliant, bloodsoaked plan to get out of his holding facility, showing both his mental and combat prowess in the process.
Ace Combat The Equestrian War rightfully introduces Mobius as a One Pony Army. He dispatches three griffon soldiers, each with a single hit and even stares down Razor. The entire Mirage squadron, especially Rainbow Dash, watches in awe as he performs his feats. By the end of the chapter, it is revealed he is a living legend and the best pegasus flier in the world.
Claymade's sequel to The Dark Lords Of Nerima introduces the new Big Bad by having him casually defeating an assassin trying to kill him (that he had hired himself), who had already easily torn through his security forces.
Hisa, in her first scene, opposes Anzu's threatening to expel Miho in order to get her to join the tankery team, showing that while she strongly desires for Oarai's tankery team to be restarted and for Oarai itself to stay open, there are limits to how far she will go.
Like in canon, Erika insults Miho, claims Oarai should not even be entering the tournament if it can't hope to win, and gets offended when Mako (Reizei) brings up the possibility of them being defeated.
Maho, seeing Miho in the cafe, is forced to conceal her emotions and choose her words carefully as Nishizumi heiress. The only thing she can bring herself to say is "I didn't think you'd still be doing tankery" After seeing that this answer upsets Miho, she thinks, "Forgive me, Miho...". She subconsciously agrees with Hisa's arguing that one should not lose sight of one's reasons for wanting to win, and as she walks off, hopes Miho is happier at Oarai.
Teru, in response to Erika derisively bringing up Hisa's being a no-show at the middle school tankery tournament three years ago, tells her, "Some people have parts of their past that they don't want to talk about." When Saki notices her and calls out to her, asking why she is denying having a sister, Teru freezes in surprise before walking off without saying anything to Saki.
In a flashback, Takako is first shown slapping a student for making a minor mistake, then defending her actions to Shiho, saying that she uses whatever means are necessary to help them improve, showing her approach to the Drill Sergeant Nasty trope.
Masae, head of the Atago School of Tankery, watches her daughters, pleased with their performance, and briefly mentions her school's philosophy. This gives some idea of how she's a less demanding parent and a less ruthless tanker than Shiho.
Hiroe, Masae's elder daughter, after winning a match, alternates between bragging about her victory and sincerely complimenting her defeated opponents for their efforts, before gladly agreeing to sign an autograph for a fan from said opposing team, showcasing that she's outspoken and full of herself, but a kind person at heart.
Being a crossover between two popular series (Ranma ˝ and Sailor Moon), Fist Of The Moon has characters that the audience already knows, but their establishing moments for each other are quite potent.
Ranma grabs three kids that the senshi were protecting from youma and get them out of danger, then comes back to help again. During this s/he takes two hits that leave her literally dead on her feet, and require the Silver crystal to heal.
Usagi: “We know her, Mars. She rescued three children and two senshi, putting her life on the line. We know her very well.”
The above quote also serves as Usagi’s, justifying (potentially life-threatening) use of the Silver Crystal for someone they only just met.
And in the same event, Rei begins by begging her not to risk her life like this, and declares her hatred of “that life-sucking rock”.
When Usa shows up for the first time, her hair is in a braid and she tries a physical assault against Usagi.
The first lines introducing Kaji in the Honor Trip Arcos Special Chapter have him scowling, & annoyed at having to wait. The next paragraph is a flashback detailing the brutalNo-Holds-Barred Beatdown he gave to a fellow soldier. Similarly, the other Arcosians are introduced in ways that reflect their personalities.
Lampshaded in Happy Riniversary, which opens with Rin wondering what to give her lovers for their first anniversary- because it's not enough for the gift to simply be good, it has to be perfect, and it certainly must be at least equal to the gift they're giving her.
...And if Rin didn't do something equally special, she would lose at the relationship.
(At this point, it is best to take a brief aside to mention the fact that Rin Tohsaka thought you could "lose" at a relationship explained a lot about what was going to happen next. At the very least, it should tell you something very significant about the way her mind worked.)
In What About Witch Queen?, Prince Ferdinand is established as a good counterpart to Hans and a honorable man when he apologizes to Anna for his brother's actions. The same scene, moments earlier, also sets him up as borderline Adorkable with his inability to form the apologies coherently:
Ferdinand: Princess, I… Well, I believe one of us, von Schwalbes, that is, should finally say that to you, because, well, it's a disgrace that it's been over three months and we remained silent, that is, I understand Friedrich, that is the king, I mean His Majesty, had his reasons and that politics stand in his way, because not everybody would be happy about it, it being what I'm about to say, that is… Verdammt. (...) I'm rambling, am I not? (he proceeds to apologize properly)
In The Wizard in the Shadows, Harry introduces himself by chasing off the Nazgul at Weathertop with a patronus, setting one on fire, and then, a little later on, when asked what he's doing by a thoroughly suspicious Merry, says a line that cements him as a Deadpan SnarkerHurting Hero and a total Badass:
"Looking for trouble."
In Prince Of The Dark Kingdom the first thing Voldemort gets to do on-screen is to crucio an 11 year old Harry for something, that technically wasn't his fault and usually would have warranted a laugh, instead of a curse...
Sunny Breeze, the protagonist of Racer and the Geek is introduced all alone on a Friday night, when he is drunk, working on a conlang, and keeping an FN FAL within easy reach.
In The Non-Bronyverse, TD's establishing moment is when he walks up with quiet dignity and grace and punches Celestia in the snout.
If the Apple Trust from the Lunaverse was not meant to be seen as unsympathetic, neigh (sic), villainous, the self-righteous and jealously monopolising behaviour they demonstrated when first appearing in Boast Busted and Longest Night, Longest Day did not help.
In Untouchable, Pyrrha tries and fails to sing. She sadly reflects that she doesn't know anything about music, as "combat had been all she'd ever known." The scene sums up a lot about hercharacter.
Jaune is introduced several sentences later, when he comforts Pyrrha and makes her laugh (at his expense) despite that.
Hypothermia precedes Weiss's actual introduction into the story with a description of Ruby, the protagonist, gawking at her beauty. And then Ruby notices that all of Weiss's admirers keep their distance from her, and that nobody sits with her at lunch. Weiss herself seems completely unruffled by this. All of this is established before Ruby has so much as a conversation with her.