Accidental Aesop: You must acknowledge your mistakes and strive to not repeat them. Sweeping them under the rug and treating them as if nothing happened is asking for History Repeats. One of the underlying causes as to why the Moonlight Butterfly keeps happening is that it destroyed everything humanity has built to the extent that they can't remember their mistakes. Since they never learned, they just kept repeating history over and over because each and every time a mistake happened it was thought to be the "first time".
Complete Monster: Gym Ghingnham is a cunning, brutal warlord. His actions range from threatening to rape his guest/prisoner Kihel Heim, to attempting to restart the Dark History, the centuries of apocalyptic warfare that nearly brought about the extinction of humanity, because he thinks it'd be fun. Gym is also awful to his subordinates. He casually murders them on the slightest pretext—up to and including an impromptu weapons test for his new suit. In one notable case, Gym kills an associate for offering him assistance in a battle—only to then turn around and beg for help from another subordinate the moment the fight is going against him.
Canonically all Gundamtimelines end with Turn A Gundam (even the shows made after Turn A). In practice, most fans disregard this and consider Turn A its own continuity, given that it doesn't really make a whole lot of sense for most Gundam timelines to mesh (or at the very most, only the timeline sequels of the UC, Future Century, After Colony and After War timelines).
Fridge Brilliance: The slight difference in Dianna and Kihel's skin tone. For most of the series it appears to just be a visual cue for the viewer to be able to tell who is who as they exchange roles back and forth, but later it's explained that multiple century-long sessions of cryosleep have drained Dianna's life force, which is why her complexion is more pale than Kihel's.
Minor villain Teteth Helleh gets some humanization just before she dies. 30 episodes later, we see the reason for her humanization (her mother) come out of Hibernation looking for her. Two characters nearby comment that they've heard the name she's calling out for before.
The Reveal about the Black History means that the struggles in every other Gundam series you've watched ended with the Moonlight Butterfly centuries later. It's worse if the series had an idealistic ending, because it means that all their struggles and hope for the future just ended with humanity falling back into devastating wars.
Pretty much how the Moonlight Butterfly works... or to be more precise, how it doesn't work when you think about it. After a set period of time, it will forcibly regress humanity back to the stone age using Gray Goo. However, humanity needs to learn from their mistakes to avoid repeating them. You can't learn from mistakes if there is nothing to learn from. And much like how Aeolia Schenberg points out, to truly escape the cycle of war, you need to change how humanitythinks as well, not just simply regress them back to the stone age and call it a day. For an analogy, the whole scenario is like giving a serial killer amnesia and taking away all his weapons. You only fixed the outside issues and symptoms, not the cause. With this system, humanity is effectively trapped in a cycle of eternal recurrent regression, forever doomed to the confines of the solar system and never truly advancing as a society. The best course of action is to hurl the Turn Gundams to the sun as soon as possible so that humanity doesn't have to worry about the Moonlight Butterfly regressing their progress, as the Black History is now public knowledge and humanity can finally learn from their mistakes and advance.
Anything with Bruno and Jacop after their HeelFace Turn, particularly when Jacop tries on Harry's glasses.
In Episode 22, a disguised Harry confronts Loran about his disguise as Laura. When Loran replies that it isn't exactly a disguise, Harry mistakes the reply that Loran means that he likes cross-dressing...
Fran also takes the piss out of the dramatic entrance of the Turn-X by noting that a turned X is still an X.
In the last episode Lily Borjarno finally gets fed up with Guin calling Loran "Laura", and suggests he put on a skirt and chase after Loran. Guin notes that society hasn't progressed to the point where someone wearing a skirt can rule the world. Mere minutes later, he is forced to flee into exile and Lily declares that she will take over control of America Ameria in her skirt.
It Was His Sled: The Reveal that the Black History refers to the events of previous Gundam series was a major plot twist; nowadays, just try to discuss Turn A without it coming up at some point.
Les Yay: Sometimes between Sochie and Miashie, Kihel Heim and Dianna Soreil, and Kihel Heim and Lily Borjarno.
The use of the Turn A to transport livestock, to the point that its Master Grade model kit includes a 1/100 scale cow.
"Master Grade 1/100 Cow, with bonus Turn-A Gundam included"
The surreally oddly well-pronounced "TURN A. GUNDAM" (by a guy people would swear was English or Australian) gets a bunch of jokes, too.
"Loran won by mashing the A button." A reference to Loran's final sword fight with Gym in which he simply spammed the same attack over and over again leading people to claim that he is Button Mashing.
Narm Charm: Harry's red, insect-eye-like shades, along with everything he chooses to wear when not in uniform.
Older Than They Think: Before the Turn A, a mustachioed mobile suit appeared earlier in the form of Butler Bensouman and before the Moonlight Butterfly, the Dragon Gundam had the Meteor Butterfly Sword.
Signature Scene: Loran using the chest compartments on the powerful Gundam to transport cows for a farmer. It's such a perfect encapsulation of the series' tone that the master-grade Gunpla includes a tiny cow figurine.
The absolutely beautiful ending in the final episode on its own, and especially if you've seen the entire series. Here it is for those of you willing to see it. Spoilers abound...it IS the ending.
The funeral of Dylan Heim in episode three. Sochie and the two house servants are absolutely overcome with grief, as is Loran over the death of a nice man who really did nothing to deserve it. To make matters worse, he is one of so many casualties that they have to rush to bury him right away, even before contacting his wife and daughter.
They Changed It, Now It Sucks!: The design of the Turn A Gundam, by American Syd Mead was very different from the standard Gundam look, and a lot of fans didn't like it; even Tomino himself has mentioned he wished he'd been more involved in reviewing Mead's designs.
They Wasted a Perfectly Good Plot: A surprising number of fans wish that Gym Ghingham's plan to restart the Dark History had been enacted, if only because of the possibility of seeing the various Gundams battle against the Turn A (although as noted under "Not His Sled", this does happen in the Super Robot Wars games). That being said, many fans did agree that the ending that the series has is one of the best.
Values Dissonance: In episode 36, Harry Ord points out that were Michael and the others who attempted a mutiny members of the Moonrace they would have been executed for their actions, noting this to be customary for a mutiny and mentioning that it is a capital offense after all. Loran and Sochie are fairly shocked by this. Falls under Truth in Television however, as a common punishment for mutiny is death.
While the series wasn't very successful with its initial ratings, it's over time grown in popularity and is these days seen as one of Tomino's best Gundam works. People are still spilt on the reveal of the Black History.
The titular mobile suit itself seems to be a walking definition of this trope. The machine have an almost uncanny ability to grow on people. Many find it to look ridiculous upon first exposure only to have it slowly becoming much more endearing in peoples eyes over time.
Wangst: Poe Eiji whose constant whining over even the slightest thing starts to get grating even to some characters in universe after a time. There is even an episode titled "Crybaby Poe" hanging a lampshade at this.
Woolseyism: In the final episode, Lily snaps at Guin that he should just put on a skirt to chase after Loran, and his reply is that Ameria wouldn't accept someone in a skirt as a leader. The fan translation implies that anyone in a skirt would be unacceptable, but the official one is more that they wouldn't want him in a skirt.