Toph Beifong's relationships

Aang
Aang has a vision of Toph in the episode "The Swamp," but it is not until the episode "The Blind Bandit" that the two actually meet. First encountering one another at the underground Earth Rumble VI earthbending tournament, Toph considers Aang a rival at first when he almost effortlessly defeats her through Airbending. At first, she is unwilling to talk to Aang and tries forcefully to keep him quiet when he talks to her parents at dinner about her earthbending skills. After dinner, however, a truce is made, and Toph discusses with Aang her ability to listen to the earth and her life under her overbearing parents. Later, Toph single-handedly saves Aang when he is held captive by the participants of the earthbending tournament. She has since joined him on his quest after running away from home, but not before throwing him into a tree via earthbending to even the score from the tournament.

In the episode "Bitter Work," Toph begins to teach Aang earthbending, but complications arise between Toph's teaching style and Aang's natural tendencies as an Airbender. Earthbending is, according to Toph, about directly taking on a challenge, not trying to find a way around or over it. The two also clash personality-wise as Aang's pacifistic, carefree attitude is ill prepared to deal with the tough and stubborn Toph. She even goes as far as intentionally acting mean by taking Aang's staff and using it as a nutcracker to get a rise out of the passive Aang. Counterproductively, Aang's peaceful nature prevents him from acting, and depression comes as a result. Only after an insightful talk with Katara about earth being his natural opposite element, does Aang start thinking positively.

Later, when forced to confront a ferocious Saber-tooth Moose-lion in order to protect a trapped Sokka, Aang finally stands his ground and learns to think like an Earthbender. Toph is revealed to have been observing all the while, and after Aang proves his new toughness by taking back his staff and moving a boulder, Toph finally proclaims him an Earthbender.

Toph often refers to Aang as "Twinkle Toes," which is likely out of friendly humor, rather than insult. The nickname originates from Aang's hovering when they fought in "The Blind Bandit." She commented then that he was "light on his feet" and asked if his tournament name was "The Fancy Dancer." In "City of Walls and Secrets" she is able to find Aang in a large crowd because of his light foot steps.

Aang and Toph's relationship has been quite strained at times. Toph's hardheaded nature clashes with Aang's on several occasions, particularly over the subject of Appa. In the episode "The Chase," a tired Toph, weary of going days without sleep, blames Appa's shedding as the reason they cannot shake their mysterious pursuers. Although she is right, Aang immediately jumps to Appa's defense, which results in Toph's abandoning the group for a short time. Later, in the episode "The Desert," after Appa is stolen by Sandbenders, Aang accuses Toph of allowing the thieves to steal him. Only Katara is able to reason with him, reminding Aang that Toph had still managed to save the rest of the group even if it was at the cost of losing Appa. Nonetheless, they have been able to work together when dealing with threats, such as the Fire Nation and the Dai Li, they appear to get along much better. In the episode "The Earth King," Aang refers to everyone, including Toph, as his family.

Katara
Katara has shown great sympathy for Toph over her restricted life under her parents. Like Aang and Sokka, Katara gladly welcomes Toph into their group. The two have been at odds with one another after Toph refused to help set up their campsite, Toph saying she can carry her own weight. Toph at one point referred insultingly to Katara as "Sugar Queen". However, at the end of the episode "The Chase," they have reached better terms with each other. In the episode "Bitter Work," Katara tries to help Toph by offering her wisdom on how to teach Aang earthbending, although Toph naturally ignores the suggestion, realizing the fundamental difference between the head-on attitude of earthbending and the redirection of Waterbending. At the very least, Toph seems to respect Katara as an equal, Bender to Bender.

In "The Desert," Katara is the only one who sticks up for Toph when Aang blames her for not saving Appa.

Later, in "The Tales of Ba Sing Se" Katara and Toph get makeovers together, and the usually self-confident Toph feels down about her looks after a group of pompous girls make rude comments about her. Katara manages to cheer her up, for which she is rewarded with a friendly (and unintentionally painful) punch in the arm. The two of them definitely respect each other and seem to be becoming good friends.

The two conflict in "The Runaway", where Toph accuses Katara of being over-bearing and "motherly". The pair make up when Katara discovers Toph's hate of her parents to truly be guilt for leaving without telling them. She asks Katara to help her write a letter to her parents at the end of the episode.

Sokka
Because of Toph's young age and blindness, Sokka is skeptical of her earthbending powers at first. Upon witnessing her skills, however, he is astonished to the point of fainting. Afterwards, he gladly welcomes Toph into their group. Just as Sokka was doubtful of Toph's fighting skills, Toph thinks little of Sokka's abilities as a warrior, dismissing his fighting skills because he cannot bend.

Though they were initially hostile, in "The Chase" Sokka is the only member of the group who does not confront Toph for her apparent lack of responsibility, and later confronts Katara and Aang with the realization that they "were pretty much jerks." The two have come to an easy understanding, and frequently work together to accomplish tasks outside the range of one or the other.

During the episode "The Serpent's Pass," Toph was knocked into a large body of water while crossing it on an ice bridge. Unable to swim, Toph called for help and, though Sokka began to jump to her rescue, Suki was able to reach Toph before Sokka had entered the water. Unaware that it was Suki, Toph gleefully thanked Sokka for the rescue and kissed Suki on the cheek. She was very embarrassed to find out what actually happened, commenting that Suki could let her drown now. The creators confirmed that the scene was intended to show Toph's romantic interest in Sokka.

When Sokka briefly left the group to pursue the tutelage of a renowned sword master in "Sokka's Master," Aang, Katara, and Toph all expressed regret over his absence, reflecting that his humor and strict adherence to a planned schedule had given them entertainment and focus in their lives. When Sokka returns, Toph claims to have not missed him, but turns aside and blushes afterwards. Sokka also seems to connect with Toph more than any other character in the group. As they depart, Sokka gives her a piece of meteorite that was used to craft his sword. Being composed of earth and metal, Toph is able to mold the fragment into any shape or form she desires, and now wears it as an arm bracelet. She later expresses how valuable her meteorite arm bracelet is to her when she let’s herself get captured by Fire Nation soldiers, only to have her plan backfire on her, and when Katara asks if she has the bracelet so that they might escape, Toph tells her she left it at camp for fear of it getting taken away from her by the soldiers. In " The Runaway" Sokka remains the only person who managed to settle the argument between Toph and Katara. He also trusts Toph enough to tell her that he can't remember his mother's face, which he hasn't told anyone before.

During the search for Fire Lord Ozai in the "Day of Black Sun", Toph uses her expert earthbending and metalbending abilities to locate the underground bunkers. Sokka constantly expresses his admiration for Toph's abilities during this search.

Iroh
Toph meets Iroh in an Earth Kingdom forest when she temporarily separates from Aang, Sokka, and Katara in the episode "The Chase." The two sit around a fire that Iroh uses to make tea and engage in conversation, although neither indicates his/her purpose for traveling or the identities of his/her relations. Iroh notes that Toph is a lot like his nephew, because they both try to do too much by themselves. He also comments that there is nothing wrong with needing help, and sometimes people offer help because they want to, not because they believe the other person is incapable. Toph thanks him for the advice, and suggests that Iroh tell his nephew how much he means to him.

Toph and Iroh encounter each other again later in the episode when the two participate with Aang, Sokka, Katara, and Zuko in cornering Azula in an abandoned Earth Kingdom town. Iroh's attention is distracted off of Azula when he notices Toph's presence and her association with Aang, Sokka, and Katara. Taking advantage of his lack of focus on her, Azula wounds Iroh with a blast of fire before successfully fleeing. Toph shows particular concern for her new friend, but leaves him after Zuko violently demands that she and the others leave him and Iroh alone.

In "The Crossroads of Destiny," when Iroh seeks Aang, Sokka, and Toph in order to seek their help in rescuing Zuko, Toph refers to him as an old friend.

Appa
Toph hates and fears flying, as it renders her truly blind; at best, she's bored and restless while flying on Appa. This is emphasized in "The Library" when she claims to have spotted the titular library from Appa's back, only to remind her friends that she is blind.

Toph expresses some dislike towards Appa in "The Chase" when she angrily points out that Appa's shedding left a large trail that allowed Azula to find them no matter how far or fast they flew.

In "The Library" Toph tries her best to protect Appa from being stolen by Sandbenders. She can't fight her hardest, as she also has to keep Aang, Katara and Sokka's only exit from an ancient library from falling into sand, which curtails her abilities. This causes her to fail in protecting Appa, and her sadness afterwards shows that she does care for Appa as a friend.