Ty Lee's relationships

Mai
Unlike Ty Lee, Mai is a lot colder and calmer. Ty Lee seemed to have met Mai when she was at Royal Fire Academy for Girls. She knew her enough to play a trick on her and Zuko with Azula and not worry about it damaging their friendship. Ty Lee occasionally teases Mai for her crush on Zuko, a romance she in fact encourages. Mai seems to be better friends with Ty Lee than Azula, because when Azula is not around, Mai doesn't follow her orders. When they exchanged stories during the episode "The Beach" and when Ty Lee told hers, Mai pointed out that she needed ten boyfriends because of she didn't get enough attention when she was a child. During "The Boiling Rock, Part 1" she and Mai went to the Boiling Rock, even though Mai went to talk to Zuko and Ty Lee followed Azula. After Mai had betrayed Azula and saved Zuko, Ty Lee blocked Azula's chi in order to save Mai. The two girls were locked away in the prison where Azula would never have to see them again.

Ty Lee and Mai presumably remained friends after their release from prison, especially in the wake of the fact that Ty Lee selflessly gave up her freedom in order to save Mai's life.

Azula
Azula and Ty lee have a somewhat complicated relationship. In some respects they're opposites. Azula is mean-spirited, intelligent, intimidating and socially inept. Ty Lee is none of those things. Ty Lee met Azula in Royal Fire Academy for Girls along with Mai. When Ty Lee completed a trick that Azula couldn't, Azula pushed her down and laughed, however, right after that, she joined in on a prank on Mai and Zuko. When Azula came to recruit her for a mission to capture Zuko and Iroh (and soon after, Aang) Ty Lee at first declined because she was happy with her life at the Fire Nation Circus. Azula then tricked her into coming along after making her show incredibly dangerous. While Ty Lee at one point regarded Azula as a friend, it is apparent that she fears Azula and is hesitant to disobey her. She seems to have more respect for Azula then Mai does. When Azula told them to follow the Avatar's friends in "The Drill", she continued the pursuit, even jumping into the slurry which Mai refused to go near. Ty Lee seems to care about Azula's opinions of her and other matters. When Azula insulted her during "The Beach", she immediately began crying. Azula apologized, showing she does care to at least some degree about Ty Lee's feelings. When Zuko called Ty Lee a "circus freak", Azula laughed. During "The Boiling Rock, Part 1", she goes to the Boiling Rock prison with Azula. She fights alongside with her until Azula finds out that Mai is helping Zuko escape. Once Mai is captured and Azula prepares to strike her down out of rage, Ty Lee chooses to block Azula's chi to save Mai. Azula had them locked away thus ending their friendship. Judging from the defiant look she gave Azula when she was arrested, Ty Lee abandoned her fear of Azula when she witnessed Azula for the monster that she was.

Like Mai, Ty Lee seems to have little respect and sympathy for Azula, as she did not even mention Azula after the War ended. She has instead moved on and made friends with the Kyoshi Warriors, whom she promptly joined.

Zuko
Ty Lee shows to have known Zuko since he was little. She seemed comfortable enough to play a prank on him. During The Beach Zuko calls her a "circus freak". She then starts to cry and tells them why she is a "circus freak". She then pressured him, with everybody else, to find out why he was so angry. After her and Azula were safe on another gondola in "The Boiling Rock, Part 2", she looked uneasy, possibly worried about Zuko.

It can be presumed that Ty Lee and Zuko are on friendlier terms with each other now that the War is over.