Bleach: Asguiaro Ebern and Luders Friegen. The latter served as the Mouth of Sauron, declaring war to Yamamoto and acting as the spokesman for the Malevolent Masked Men. The former is the first member of the Vandenreich that we meet, who attacks The Hero within the first five chapters of the latest arc. Luders was thought of as The Dragon; Ebern was seen as The Rival to The Hero. Both were killed by their boss shortly after they're introduced to quickly establish that their boss has an in-universe reputation for possessing a callous disregard for his subordinates.
"Hot Ice" Hilda from Outlaw Star, though it was nearly far enough to qualify her as a Sacrificial Lion. Clearly a case of being Too Awesome to Live.
One of the most infamous Lambs in all anime is Musashi Tomoe from Getter Robo. He's not nicknamed "Kenny" just for the lulz, huh. He did make it through the entire original Getter Robo before making his Heroic Sacrifice, but later versions would kill him sooner- Shin Getter Robo vs. Neo Getter Robo makes his Heroic Sacrifice the opening scene.
Not surprisingly, Mazinger Z also has a lamb. MORIMORI-HAKASEEEEEEE!!!!!
Musashi's replacement, Benkei, doesn't fare any better in the Getter Robo manga by Ken Ishikawa either. Ryoma even lampshades the fact.
Sophie Oswald (Leon's sister) and Aaron Killian (Yuri's father) from Kaleido Star.
Kazuya is the first character to die off in Mai Hi ME when his girlfriend Akane has her CHILD destroyed by Miyu, after the latter exposes herself as The Mole. Akane herself survives, but it takes her the entire rest of the series to recover from the ensuing Heroic B.S.O.D..
Higurashi: When They Cry subverts and uses this by killing Mion, Rena, Keichii, Takano and Tomitake in the first arc, only to see them back again because of the "Groundhog Day" Loop. They die again in other arcs too. Frequently. However, we find out halfway through the second season that Takano actually never died, and never actually does.
The apparent Big Bad of Code Geass, Clovis, is killed within the first few episodes. By the protagonist. And he is in no way the last.
In an interview, the show's head writer commented that Clovis didn't last as long as originally planned, while the Ensemble Dark Horse (the one with the page image) lasted a lot longer than intended.
This is pretty much what Monica and Dorothea are there for. Bismark, too, to a lesser extent. Gotta give the audience the impression that the final battle is going to be a slaughter house.
In Ga-Rei -Zero- the entire cast is one of these, killed at the end of the first episode. Then we meet the real cast, at *their* death scene, then we go back in time.
Black God quickly knocks off one of the main character's classmates and that cute kid from next door in rapid order, both during the first episode.
Darker Than Black has a serious case of this, setting up a ludicrously obvious Meet Cute and then smashing the whole thing to tiny little bits at the end of the second episode. That lady who looked like a love interest? The one who everyone was looking for and who knew a lot about the Gate? She was actually a doll programmed with the original's personality as a trap for Hei, and the main reason for his repeated rescues (not to mention flirting) was so that he could get information out of her. The original Chiaki was Dead All Along. By the end of the second episode, Chiaki has had this happen to her twice. It says a lot about the series that someone so obviously set up as a major character had been used as a human shield within two episodes of their introduction.
Nika plays this role in the second season. He's a friend and possible crush for Suo, not to mention the other person among their friends to suffer from becoming an Un-person. Not to mention the possible drama of him interacting with friend-turned-Contractor, Tanya. Nope, he's just horribly murdered by Tanya, which serves the purpose of unleashing Suo's Superpowered Evil Side.
Subverted with Suo's dad, who also gets killed off very shortly into the second season — the one Hei killed was actually a copy made by Shion's power, and the original escaped.
In Shikkoku no Hana manga a girl who just became a really interesting character was blasted into dust simply by a freakish accident no one wanted or expected, leaving Hei and her bewildered friend only a shoe.
Kotori from X1999 dies early on. It is foreordained that she will die.
In Naruto, this role belongs to Ill Boy Jonin Hayate Gekko, the original proctor for the Third Stage of the Chunin Exams - he gets a little screen time (in which, to be fair, he does a fair bit) before being the first named good guy from the main storyline to snuff it. It's true he dies later than most, but still qualifies as his death heralds both the steadily darkening storyline and the first major character death, 3rd Hokage Hiruzen Sarutobi.
Kanade from Senki Zesshou Symphogear. Symphogear actually wasn't very kill-happy at all, but they used Kanade's death in the first episode to plant that little seed of doubt in the audience's mind that character X might be dead for real.
Monster has quite a few of these, such as Mauler and Nina's foster parents, but their deaths are legitimately tragic, and they're remembered by the other characters rather than simply forgotten.
Everyone not named Yui in the first two episodes of Muv-Luv Alternative: Total Eclipse. And considering those characters were filling out standard character role tropes nicely... the notable Bait-and-Switch nature of the anime kicks in. Horrifically.
Psycho-Pass: Yuki, a minor character who was introduced as Akane's friend outside the Public Safety Bureau early on, is used to lure Shinya in one of Senguji and Makishima's "hunting games". While she and Shinya manage to kill Senguji, Makishima captures her and shortly after kills her in front of Akane as punishment for the latter's inability to shoot him. This is the first time Makishima kills somebody with his own hands, rather than indirectly causing their death.
Nina Tucker in Fullmetal Alchemist - or, rather, the chimera that was created out of her and her pet dog by her Mad Scientist father. It's Played With a bit in that her death does have a big effect on Ed and Al's Character Development and they bring it up often later (usually to reflect on how they were powerless to save her) - but it doesn't do much to move the plot other than to confirm what the audience already knows about how screwed up the Amestrian military's priorities are. Overall, her death contrasted with that of Hughes later on is a good way to illustrate the difference between this trope and Sacrificial Lion.
Baccano! kicks off its three-way train hijacking when a cultist shoots the Flying Pussyfoot's cheerful redheaded young conductor at the Deadly Closing Credits. Or so it seems. The gunshot is for the cultist's death, after said young conductor kicks the gun out of his hand and turns it on him. Then the young conductor makes his "corpse" by nabbing a second hijacker and shaving his face off with railroad tracks. Then the young conductor decides he's going to deal with this whole train hijacking situation by going on a blood-soaked killing spree. And thus the Rail Tracer is born. For bonus points, he even hangs a lampshade.
Attack on Titan: Carla Yeager, Eren's mother. She gets devoured by a Titan in the first episode to motivate Eren to kill all the Titans and to show how violent the series is.
In the 3rd episode of the animated Adventures of Maya the Honeybee, a dragonfly eats a fly just after Maya gets to know the fly. It is maybe the only named character to die.
The anime of Magic Knight Rayearth turns Presea into this. She's featured prominently in the opening and closing credits and starts the same way she did in the manga: a quirky blacksmith who becomes a Cool Big Sis to the girls, with an added scene where she fights off some monsters. In the sixth episode, Ascot and his creatures destroy her house and she's killed in the collapse, underlining how high the stakes are. (Her Death by Adaptation isn't forgotten and more ramifications happen in the second season.)
Tomonori Komori from Narutaru. When he's first introduced it seems like he'll be an important part of the plot and is fairly intimidating, only to get messilykilled off a few chapters later. His death sets the tone for the rest of the series.
Mami Tomoe of Puella Magi Madoka Magica is the first Magical Girl encountered by the protagonists in the series (if you don't count Madoka's dream in the very first episode). She spends the first two episodes being the resident Cool Big Sis to the rest of the girls and a possible Big Sister Mentor for Madoka before being killed off in gruesome fashion at the hands of Charlotte, Episode 3's Witch of the Week, which does an excellent job at establishing just how dark and messed up a series it is. Mami is not forgotten, and later episodes show previous timelines of Homura's "Groundhog Day" Loop where Mami survives her encounter with Charlotte, only to meet some other bad fate — one timeline has her hitting the Despair Event Horizon due to Sayaka becoming a witch and having to be put down, and Mami realizing that this is the fate of all Magical Girls, which drives her to try to kill everyone and herself to spare them from this horrible fate, killing Kyouko and almost killing Homura before being stopped by Madoka. Mami, along with every other Magical Girl, ultimately comes back after Madoka's wish destroys all witches before they are born, though this comes at the cost of Madoka's own existence.