NOTE: You may put examples from both the show and the graphic novels.
A typical reaction from a fan who notices such idiocy.
"Imprisoned" introduces us to a large naval vessel used by the Fire Nation as a prison for Earthbenders. The Earthbenders are unable to fight their way out due to the ship being made entirely out of metal, and the fact that metalbending has yet to be invented. So far, pretty smart, but the fuel they use in the ship's furnaces ruins it. You'd Expect: The Firebenders to use wood, or any other substance that can't be bent by an Earthbender. Instead: They use coal as their fuel source. For a prison dedicated to holding people with the power to manipulate earth and stone. Granted, they presumably keep the existence of the coal secret from the prisoners, and the prison does a pretty good job of breaking their spirits, to the point that even when given earth and stone, it takes them a while to start fighting back. But with that being said, the prison still creates its own Achilles' Heel by having coal there. Note: These Firebenders are still geniuses compared to their film counterparts, who imprisoned the Earthbenders in a natural valley made entirely of earth. Not that it excuses their moment here.
Season 1 finale. Pakku catches Aang teaching Katara behind his back, against Pakku's wishes. You'd think: With the fate of the world on the line, he'd at the very least let Aang off with a warning, or maybe pretend he didn't see them, if not drop the "no-females" rule altogether. Instead: He refuses to teach Aang further. Makes you wonder why Roku didn't make a reappearance when he appeared to Jeong-Jeong over a far more trivial refusal. Note: By the next morning, he is willing to reconsider, provided Katara apologize. He likely only refused to keep up his smugness over antagonizing Katara.
Concurrently with the above: Aang and Katara have to apologize to Pakku. He makes a smartass remark. You'd think: Katara would let it slide. Again, fate of the world. Not really the time to get angry. Instead: She starts talking shit right back, challenges him to a duel and then attacks him with he refuses, any of which would ensure he'll never agree to teach Aang again.
And of course, the defining idiocy of the series comes from Admiral Zhao during the first season finale. Zhao is invading the Northern Water Tribe in order to stop Aang from learning waterbending. Zhao is also aware that the Ocean and Moon spirits (the entities that power waterbending) are in the heart of the Tribe's city in the form of relatively harmless koi fish. Killing the Moon Spirit would destroy waterbending, however it would also cause serious trouble for the whole world, as both Aang and Iroh warn Zhao. You'd Expect: Zhao to realize that killing the moon is not only a mite bit overkill, but also would hurt the Fire Nation just as much as the rest of the world. So he'd think of some alternative that isn't nearly as stupid. Instead: Zhao kills the moon. Brilliant. And it's pretty clear that to him, the destruction of waterbending was a bonus. He just wanted to be able to brag that he killed the moon. Fortunately, Yue is able to undo his actions with a Heroic Sacrifice. Result: Aang fuses with the now extremely pissed off Ocean spirit, transforming the both of them into a gigantic humanoid koi monster. Koizilla proceeds to lay waste to the entire Fire Nation force, preventing Zhao's conquest of the Northern Water Tribe, and Zhao himself is later dragged underwater by the Ocean spirit. He has a cameo in The Legend of Korra, where it's revealed that he's ended up in the Fog of Lost Souls as punishment for his actions, condemning him to eternal insanity.
Following Zhao's unsuccessful invasion of the Northern Water Tribe, Azula is assigned to bring Zuko and Iroh back to the Fire Nation, the former because of his failure to capture the Avatar, and the latter because of his betrayal at the North Pole. To ensure the pair co-operate with her, Azula lies to them about the Fire Lord seeing the value of family and wanting Zuko back. You'd Expect: Azula to have drilled into her troops the importance of upholding the façade until Zuko and Iroh are safely chained up and muzzled. And if she did that, the troops to do so with the utmost dilligence until Azula orders them otherwise. Instead: One of Azula's officers refers to Zuko and Iroh as "the prisoners", leading to the pair of them realizing it's a trap and inflicting a Curb-Stomp Battle on the crew before making their escape.
During one of their trips through the Earth Kingdom, we had a crazy general trying to force Aang into the Avatar State. You'd Expect: The General would NOT try to piss off the AVATAR. Or to just realize that you CANNOT control a supernatural force. Instead: He has his men attack Aang for twenty minutes of the episode, and when he finally does go into the Avatar State, the crazy bastard doesn't seem to mind that he and his men are getting their ASSES WHOOPED. Even when Aang's finally finished his mad rampage, he wonders out loud how to control him when he's in that state. Thank you for interfering, Sokka.
While wandering the Earth Kingdom in the beginning of season 2, Iroh comes across a plant that is either "the rare White Dragon Bush, whose leaves make a tea so delicious it's heartbreaking!", or else "the White Jade Bush, which is poisonous." You'd Expect: He'd demonstrate the good sense possessed by even the most scatterbrained Cub Scout and not touch the thing with a ten-foot-pole until he's 100% certain it's the right one. Instead:He drinks it and almost dies.
Wan Shi Tong has let the team into his library, one of the most sacred things he has. He is incredibly suspicious of them and all humans, due to his belief that humans only seek knowledge to compete with other humans, and the fact that the last human visitor burnt down a section of his library and used his knowledge for destruction and violence. He tells them not to do the same as him, however Sokka finds some partly destroyed information which he can use to find a fire-nation weakness. You'd Expect: They'd use the utmost discretion, continue to check for his presence, not only out of respect for abusing the library, but also because Wan Shi Tong is a spirit (who dwarfs them), and the last angered spirit took on an entire armada with the help of the Avatar. Instead: Sokka quite loudly announces that with the information they can finally destroy the Fire Nation, and Wan Shi Tong is obviously right behind him. Wan Shi Tong subsequently tries to kill the Gaang and bury the library for good, with the lot of them still inside it.
Iroh has just made it into the heavily-defended Earth Kingdom city of Ba Sing Se alongside a boatload of Earth Kingdom refugees, who would not look kindly upon a Fire National sneaking in with them. He has just gotten a cup of too-cool tea. You'd Expect: Iroh would just throw the tea away and wait to get something better, or simply drink it up and bear it. Or: Iroh wouldn't make such a fuss about the tea being cold and very quietly heat it up before drinking it. Instead: He makes a loud fuss about the tea being cold, and THEN uses Firebending to heat the tea up, putting himself (and, by extension, his nephew) in danger of being lynched as Fire Nation spies. And, in fact, he does get noticed by Jet; while things don't exactly go as Jet planned, he still comes very close to outing them as Firebenders.
Long Feng, the Evil Chancellor of the Earth Kingdom capital, has a problem with the Avatar and his companions running around the city talking about the war, which he has gone to great lengths to deny so as to keep himself in power. Also, they have crucial intel on the Fire Nation. You'd Expect: Long Feng to arrange a meeting immediately, listen to their plans/intel, give them the "no war within the walls" speech, then either direct them to one of the outer wall generals or just politely ask them to leave. He gets what he wants and the Gaang gets what they want, at least to some extent. Most importantly, his rule is not threatened. Instead: Not only does he actively try to stop them from sharing their intel, but he blows them off when they finally do meet face to face. As Aang is already insistent on staying in Ba Sing Se to look for Appa, this only further antagonizes them until the season finale hits and he's deposed. Worse Still: Long Feng captures Appa partway through this. He literally has a trump card that will either get rid of the Avatar or at least make him cooperative. You'd Expect: Him to either threaten Appa directly to force their cooperation, or quietly have Appa moved outside the city walls and point them that way. Instead: He keeps Appa secret for no good reason and tries to send them on a wild goose chase with Jet, which could have been accomplished just as easily by letting Appa go.
Zuko does a few less than brilliant things, but consider the time he finds out that Aang in in Ba Sing Se, too, thanks to the "lost pet" flyer about Appa. You'd Expect: If Zuko really wants to take up chasing Aang again, he make use of Aang's address written on the flyer. Instead: He infiltrates the base of the local Secret Police, which is made up mostly of elite Earthbenders, planning to steal a huge, flying furry monster which has horns, airbending, and no reason to be cooperative, what with having defended Aang against Zuko before. And Zuko has no idea whatsoever what he'd do if he actually got Appa out of there. It takes his Uncle coming after him and spelling out the flaws in his plan and how he's blindly following someone else's will to snap him out of it.
The newly minted Avatar Roku is celebrating his wedding, when he is approached by his best friend Fire Lord Sozin. Sozin tells Roku that it would be just swell to Take Over the World. You'd Expect: Roku to try and talk him out out it, to use the experience he gained in traveling the world for 12 years to explain why taking over the world is a bad idea, to take Sozin on a Diversity Tour of the world, to latch onto Sozin's arm and not let go until he is sure that one of the 5 most influential people in the world lets go of his megalomaniac ideas. Instead: Roku blows Sozin off and tells him to just forget it. And when Sozin implores him to listen, Roku says that he doesn't want to hear anymore of this, leaving Sozin feeling betrayed by his best friend, bitter and isolated. This war really IS your fault, Roku.
Many years ago, a Waterbender from the Southern Water Tribe by the name of Hama was captured by the Fire Nation. She eventually escaped by developing Bloodbending and using it to incapacitate her guards. As a result of the torment she suffered at their hands, Hama hates the Fire Nation and everyone in it. Meanwhile, she hopes to pass on Bloodbending to the next generation of Waterbenders. You'd Expect: Hama to try and contribute to the war effort against the Fire Nation, thereby using her hatred of them constructively, and/or seek out one of the three Water Tribes in the world, so that she'll have more of a chance of passing on her technique. Instead: Hama just sits on her ass in a Fire Nation village, and her revenge consists of her using Bloodbending to abduct a few of the villagers every full moon.* The only time she can use Bloodbending.
Hama eventually does find a potential student by accident: Katara, who is also from the Southern Water Tribe and has a reasonable grudge against the Fire Nation, since Fire Nation soldiers killed her mother. During the full moon, she reveals to Katara the art of blood-bending as well as her role in kidnapping random Innocent Bystanders at night. You'd Expect: For Katara to go Oh Crap! on realizing that she's facing a Black and White Insanity woman that can manipulate people like puppets with their bodily fluids and has done so for decades. Katara could then try to make an excuse to leave as soon as possible, create a distraction and run for the cover of the woods where Hama won't be able to see her without the moonlight, or beg to only learn the movements without having to use them on a living thing. Either way, it's best for her to go Screw This, I'm Outta Here! and find the others. Instead: Katara openly declares that she will never use blood-bending. Hama then blood-bends her, forcing her to learn the movements anyway, and lampshades Katara "betraying" her without having learned from her. Although Katara eventually overpowers Hama, the scars of blood-bending stay with her, especially when forced to use to to save Aang and her brother.
In "The Day of Black Sun" Azula suspects that Aang is alive, and lures him and his friends into an isolated room. You'd Expect: That Azula would bring a large squad of Dai Li agents to overwhelm them. Or she could have surrounded the area with firebenders, and had them attack after the eclipse was over. Granted, one could argue that resources are important elsewhere, but the Fire Nation had more than enough troops to force a surrender, and the Avatar is the most important figure in the entire war! Instead: She attacks them with only two Dai Li agents. The Gaang defeats them, and they even manage to catch Azula herself! It was basically luck that she wasn't injured, or killed. Azula does manage to escape, but so does the Gaang (again), and Aang inevitably fulfills his destiny, defeating Ozai and the Fire Nation. Oops.
In "The Western Air Temple" Zuko has decided to join Team Avatar, who are dealing with the aftermath of the failed Black Sun invasion, and follows them to the titular temple, to become Aang's firebending teacher. You'd Expect: That Zuko would show up with a peace offering, say supplies for the escapees of the Black Sun fiasco, or his father's plans for Sozin's Comet. Last time Team Avatar saw him, Zuko had helped his sister take down Ba Sing Se, turn on Katara and betray his uncle; they have no proof that he is trustworthy, especially after Azula Out-Gambitted them during the Black Sun. Instead: Zuko, being Zuko, only practices his "Hello, Zuko here" speech on a frog and while apologizing to the Gaang for his past actions reveals that he sent "Combustion Man" after them. ("That's not his name.") If not for Combustion Man arriving just in time and aid the Gaang against him some earning him some good favours, he would have had no chance of convincing them at all.
Meanwhile Aang still needs a Firebending teacher, and lo and behold, Zuko appears and basically offers assistance to the Gaang. He even offers to surrender as a prisoner if they don't trust him. You'd Expect: That Sokka and Katara put aside their reasonable grudge against Zuko, consider the fate of the world and how Aang really needs to learn Firebending and reluctantly allow Zuko to join. Keep in mind that he's quite possibly the only Firebender in the world now who would be willing to help since Jeong Jeong is in hiding and Iroh, who helped the Gaang escape in the season two finale, is presumably imprisoned. In addition, given the Gaang has three benders "and Sokka" as Toph put it once, they could all serve as Betrayal Insurance if Zuko is lying or becomes tempted to return to the Fire Nation's side, especially Katara with her Team Mom attitude and recently-gained ability to bloodbend. Instead: They pretty much tell him to get lost. Even Toph sees the stupidity in their action. Thankfully: Combustion Man arrives and Zuko helps the Gaang defeat the very same man he hired, allowing them to grudgingly let Zuko join the group.
Series finale. Aang has been advised in a two to one vote by the previous four Avatars (Yangchen and Kyoshi vs Kuruk, with Roku just saying "Be decisive") to kill Fire Lord Ozai. You'd Expect: Aang to take their advice and put aside his own spiritual needs. Don't want to pull the trigger yourself? That's fine, hand over control to the Avatar State, or one of the previous Avatars. Or: If Aang doesn't want Ozai to be killed through him, he could use his bending to restrain him either until Ozai can be taken into custody, or until Sozin's Comet passes, stripping him of his souped-up firebending. There's lots of options. Instead: He gambles the safety of the world on a completely new, totally untested technique that he learned like a day ago, and very nearly loses his life in the process. It works (barely), but it was an unacceptable risk with no inherent benefit.
The Promise: Following Ozai's defeat and the subsequent end of the Hundred Year War, the Harmony Restoration Movement is established to remove the Fire Nation's colonies in the Earth Kingdom. One year later, Zuko travels to the colony of Yu Dao, where he learns that the Fire Nation colonists and the Earth Kingdom citizens have integrated deeply with one another over the last century, to the point that forcibly relocating the colonists would cause more problems than it would solve. Because of this, Zuko decides to end his support for the Harmony Restoration Movement. You'd Expect: That before taking any "drastic" action, Zuko would get in contact with Aang and/or the Earth King, explain the situation to them, and see if they could work things out peacefully. Instead: Zuko publicly withdraws his support for the Harmony Restoration Movement, orders the departing colonials to return to the colonies and fortifies Yu Dao, not letting anyone in or out. Unsurprisingly, a lot of people see this as Zuko refusing to give up his colonies, and another war nearly starts.
Aang decides to celebrate Yangchen's festival, an old Air Nomad holiday, with the rest of the Gaang and the Air Acolytes. Toph soon soon stops enjoying the holiday when Aang insists on following through with the ceremonies of the holiday despite not knowing any of their purposes, since it reminds her of her unenjoyable childhood. It gets to the point that she becomes visibly upset, and Aang notices. You'd Expect: That Toph would explain her issues with participating in the holiday to Aang, and either ask if she can be excused from the celebrations or just suck it up and stick it out. Aang may have been a bit overly-enthusiastic about celebrating Yangchen's Festival with his friends, but it's not like he chained her up and dragged her along. Alternatively, Aang could just ask Toph what her problem is. Instead: Toph pretty much decides to behave like a spoiled brat, whereas Aang immediately treats her like such without ever thinking that she might have a reason for her behaviour.
As Aang prepares to destroy the Earthern Fire Refinery and the surrounding town in order to try and pacify General Old Iron, Toph appears and demands to know what's going on, being one of the biggest fans of the refinery and all it represents. Aang now has to give her an explanation. You'd Expect: "There's a giant, all-powerful spirit who's going to go to war with humanity if this land isn't returned to its natural state!" Instead:(paraphrased) "One of the previous Avatars made a deal with a giant, ancient spirit, then humans built over this land and the spirit's probably coming back now...you know what, forget it! I have to do this, so just trust me!" Result: This explanation completely fails to satisfy Toph, who assumes that Aang is once again putting his people's traditions over the rest of the world, and the two of them get into a battle that could have ended in one of them getting killed if Old Iron hadn't shown up when he did.