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 when 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. Not to mention that Iroh, one of the strongest firebenders in the world, had just threatened him to stop. 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, like maybe taking the moon spirit hostage to ensure the cooperation of the Northern Water Tribe. 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 war-weary general trying to force Aang into the Avatar State, to end the war with the Fire Nation early. After a Failure Montage, which includes giving Aang caffeinated tea and pouring mud on him, Aang admits that he thinks the state can only be activated when he is in danger. You'd Expect: The General would Know When to Fold 'Em and NOT try to piss off the AVATAR. Or to just realize that you CANNOT control a supernatural force if it needs a violent trigger. Instead: He has his men attack Aang in the climax, and when Aang 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.
While the Gaang is taking mini-vacations, Sokka is determined to use his for war intel to beat the Fire Nation. He finds out from a scholar that there's a great library in the desert, run by spirits. When they get there, the Librarian Wan Shi Tong asks if they are looking for war knowledge. Wan Shi Tong 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 You'd Expect: Aang, spiritual being as he is, to remember that messing with spirits for their knowledge can lead to bad things happening. You'd expect he would warn Sokka about this. Also, given a spirit kidnapped Sokka a season ago that he would remember Do Not Taunt Cthulhu. Instead: Aang rapidly vouches for Sokka when the latter asks and doesn't warn him about the danger of messing with spirits. He also doesn't remonstrate Sokka for slipping various scrolls into his sack. This means that when Wan Shi Tong catches them, Aang can only give a weak apology that they're trying to do the right thing. Wan Shi Tong then points out that all humans believe that they are doing the right thing, and he's not going to let them abuse his knowledge anymore.
Wan Shi Tong has let the team into his library, one of the most sacred things he has. Sokka finds some partly destroyed information which he can use to find a fire-nation weakness. You'd Expect: Having gone this far, they'd Do Wrong, Right and 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. Karma later hits the Gaang: the intel about the solar eclipse leads to a failed coup in season three, with most of the misfit army captured.
While talking to the scholar, he warns them about desert Sandbenders. The Sandbenders notice the Gaang at the dried up tavern, along with their flying bison. You'd Expect: The Sandbenders to realize that flying bison are rare, and thus it must belong to the Avatar, who is the Earth Kingdom's only hope of ending the war. Also that they must not reinforce the stereotypes that they are petty bandits, which the scholar mentions. Instead: The Sandbenders don't consider these ramifications. They follow the Gaang into the desert and steal Appa while they're all in the Library, stranding the Avatar in a hostile environment. If Aang had died in the desert, the Fire Nation would have won then and there. Then when the Gaang miraculously makes it out, the ringleader of the thieving Sandbenders accuses them of stealing the abandoned sandcraft, while his father points out if the Gaang survived the desert they are guests and deserve Sacred Hospitality. Toph, who witnessed Appa's kidnapping, recognizes the voices, and Aang goes into the Avatar State on hearing Appa was muzzled, destroying everything in sight. The son's weak apology that he didn't know it was the Avatar's bison does not placate Aang, and his father shouts, "What have you done?!" If not for Katara, the Sandbenders would have died by sandstorm.
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.
Jet on realizing Iroh is a firebender decides to get proof. He finds that Iroh is more resourceful thanks to Zuko chiding him, and thus Jet gets no evidence. You'd Expect: Jet to just let it go, as his friends advise him. They're starting a new life, and so far Iroh and Zuko aren't posing a threat to anyone by making tea. He can act if he sees anything suspicious. Instead: Jet storms into the tea shop and threatens Iroh at swordpoint, gambling that Iroh and Zuko can only firebend. To the patrons, he looks like a lunatic threatening a harmless tea maker. Zuko then defends his uncle with his swords, and fights Jet long enough for the police to arrest him and take him away.
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.
When Katara goes after the Fire Nation General that killed her mother, we learn that he did so during a raid to find a rumored waterbender in hiding. It turns out Katara's mother lied to the general about being the waterbender to protect Katara. You'd Expect: That the general would demand proof to ensure that they had found the right person. Instead: He instantly believes her without question and kills her, allowing Katara to remain hidden. Given that Katara was the one who ended up discovering Aang, the general's idiocy indirectly led to the downfall of the entire empire.
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. It hits a snag when it comes to the older colony of Yu Dao, where the Fire Nation colonists and the Earth Kingdom citizens have intermarried and 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. You'd Expect: The city mayor and his family to contact Zuko and explain Yu Dao's situation to him, and try and persuade him to leave Yu Dao alone. It also wouldn't hurt to remain polite and respectful during any negotiations, considering that Zuko is their freaking head of state. Instead: Kori Morishita, the mayor's daughter, attempts to assassinate Zuko, which results in her arrest, and Zuko storming over to Yu Dao to personally enforce the Harmony Restoration Movement there. While he's there, Kori and Mayor Morishita frequently belittle him, even comparing him negatively to Fire Lord Ozai, the genocidal and maniacal tyrant who abused Zuko for most of his childhood. Result: The only thing that stops Zuko from imprisoning Kori and her father for the rest of their lives is Mrs Morishita - practically the Only Sane Man in the family at this point - diffusing the situation, after which Zuko proves to be a Reasonable Authority Figure who would have been perfectly willing to help Yu Dao, if only he'd been made aware of what was going on beforehand.
As a result of the above, 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.
Shortly after Zuko does this, Aang and Katara show up in Yu Dao hoping to talk to him. A Fire Nation soldier orders them to leave, and Aang explains that he just wants to talk with Zuko. You'd Expect: That Zuko would have anticipated Aang showing up, and given his guards sensible instructions on what to do if and when he arrived. Even if Zuko didn't, there is nothing stopping the guard from simply asking Aang to just wait at the gates while he goes to find Zuko. You'd Also Expect: Zuko, who has known Aang and Katara for over a year by this point, to know that they're not going to attack his troops for no good reason. Instead: The guards attack Aang and Katara, and keep doing so even when Aang insists that he only wants to talk. Then when Zuko shows up, rather than ordering his guards to stop attacking, he tries to restrain Katara. This causes Aang to go into an out of control Avatar State, and the only thing stopping Zuko from becoming a smear on the street is Katara talking Aang out of it. And after all the above, Zuko has the gall to tell Aang that they both need to calm down and talk, even though, as Aang rightly pointed out, that was what he was trying to do in the first place.
Zuko is clearly conflicted on what to do regarding the colonies and decides to ask someone for help. You'd Expect: That he would consult his uncle, who has always given him good advice, or perhaps someone from the Gaang, who could surely help him with the whole Harmony issue. Instead: He consults the Jerkass and manipulative Ozai. While Ozai does give Zuko advice regarding leadership, the way he gives it makes Zuko reluctant to take it, for fear that he'll be acting just like his father.
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.