In Avatar: The Last Airbender, the raging war and threat of imminent destruction is all but overwhelming, and takes up most of the narrative space— as well as constantly occupying the thoughts of its main characters. And yet, the promise of new romance continued to bloom across the series, starting with Aang and Katara's meet cute in the series opener.
Over the course of three seasons/books, several others find their own version of courtship. And while not all of them survive until the end, they make the characters involved blissfully happy for the duration of the relationship's existence, not to mention softening up a series that is otherwise an action-packed and often dour affair.
Toph & Sokka
Toph and Sokka start out on a negative note, with her mocking him for not having any bending abilities. However, since only he abstains from criticizing her terrible behavior, she develops a minor crush on him.
In fact, she has even blushed while in his presence during the hunt for clues about Hama, and what greater sign is there that Toph has it bad for Sokka? Still, whatever became of this relationship is unknown— perhaps they got together in the distant future, resulting in the birth of Suyin Beifong?
Zuko & Jin
The first girl that Zuko ever takes out on a date, Jin meets him through the tea shop where he and his uncle are employees. He is adamant on looking his best for the outing, but she says that she prefers a more casual look while playfully mes his hair. Post dinner, Zuko is taken to Jin's favorite location, a gorgeous fountain surrounded by unlit lanterns.
He decides to bring her some joy by lighting them with his Firebending, while inwardly hating himself for revealing his secret. However, Jin doesn't see this in any way other than romantic, and just goes for the kiss. Of course, Zuko's not done dealing with his infinite other problems at this point, and runs back home immediately after.
Sokka & Princess Yue
Sokka finds Yue a vision of perfection right from the moment he lays eyes on her floating past in a royal gondola. She also has the same feelings, but her betrothal and related patriotism prevents her from offering herself to him. Sadly, Zhao's cowardly deed forces her to shed her human form and turn into the moon, thereby restoring global balance.
Princess Yue is Sokka's first kiss, explaining why she has an eternal place in his heart. For instance, he reacts angrily when Toph assumes that she is behind the abduction of Fire Nation townspeople.
Toph & Kanto / Toph & Satoru
In The Legend of Korra, Lin Beifong demands to know her father's identity, at which point Toph confesses that it was a "nice man" named Kanto with whom she had an amazing but short-lived relationship. While Suyin's dad has not been confirmed yet, her mother did have a fling with Satoru, who becomes the main patron of her Metalbending Academy.
Toph defends him when Aang accuses his refinery of being the main cause of the local water pollution, claiming that her seismic sense confirms this fact by perceiving no change in his pulse rate. They break up shortly after.
Katara & Jet
Fine, Jet was never a "romantic partner" for Katara, but if he had not turned out to be a backstabbing little weasel, who knows what might have happened? She is clearly infatuated with him from the moment he takes down an entire Fire Nation camp mostly by himself, which Sokka finds infuriating.
More importantly, Katara likes him so much that she is willing to ignore her brother's warnings about Jet, which obviously comes back to haunt her later. Additionally, if Jet had been a good guy the whole time, perhaps a love triangle between the three of them would've resulted.
Zuko & Mai
Zuko begins dating Mai once he obtains his precious honor, and a seat by the right hand of the Fire Lord. Both of them being taciturn creatures, one wonders how they even communicate anything to each other.
Regardless, Mai and Zuko's relationship has weathered some harsh storms, including when both of them betray Azula and have to face the consequences. They split up in the extended canon, because the new Fire Lord has retained his old habits of hiding everything from his girlfriend. This calls into question of the identity of Izumi's mother.
Suki & Sokka
Sokka's initial meeting with Suki is quite the humiliation for him, but he learns a valuable lesson that day on Kyoshi Island— that being a girl and a warrior are not mutually exclusive notions. When they team up in Ba Sing Se, he shows his affection by kissing her, which sort of establishes a tentative relationship between them.
After the Hundred Year War, Suki becomes a royal bodyguard for Zuko, but it looks like she's still "dating" Sokka as they do evidently couple things together. Their destinies, however, take different paths.
Aang & Katara
The future of Aang and Katara's relationship is palpable from the beginning, but their journeys together are what evolve an adolescent crush into a lifelong commitment. They are intensely loyal and devoted to one another, even in the face of potential death, considering that Katara emphasizes that "Aang won't lose [to Fire Lord Ozai]. He's going to come back. He has to."
After everything is said and done, they start dating properly, going through a lovey-dovey phase that Sokka finds annoying and unseemly. Following their marriage, they parent three children: Bumi, Kya, and Tenzin, who become essential aspects of The Legend of Korra narrative.