This article contains major plot spoilers from the light novels that have not been revealed in the anime. Proceed at your own risk.
Mamiko is Kumiko's older sister. She likes to tease her younger sister, and she was also the person who inspired Kumiko to play the euphonium. They appear to occasionally experience sibling arguments, but they share a close bond as sisters.
Akiko is Kumiko's mother. Little has been revealed so far in the anime about their relationship.
Kitauji High School
Kumiko and Reina have a quite complicated relationship. In junior high, Kumiko saw Reina being upset and crying because they did not advance to the nationals despite getting gold. This prompts Kumiko to absent-mindedly ask if she really thought they could have advanced. This offends Reina, thus creating tension in their relationship.
Every time they meet, Kumiko becomes very awkward, but Reina always behaves naturally and does not show any signs of embarrassment or discomfort. After a several talks with each other, they resolve their issues and grow gradually closer to the point that they both blush when they see each other. In episode 8, Kumiko invites Reina to go to the Agata Festival with her, albeit unintentionally. However, after they begin conversing and relaxing with each other, they drop the polite form of speech and begin calling each other by first names by the end of the night, signaling a new level of friendship between them.
Now that they are friends again, Kumiko is probably the closest to Reina, since they intimately hug each other and share a strong bond of trust. They both have even said to each other that they love each other. Kumiko also says that she "will always stay by Reina's side" when Reina would compete against Kaori. Kumiko said that she "would never abandon her, even if it would cost her life." This indicates that the two share a amicable relationship that is on higher level from Kumiko's other friendships.
Kumiko seems to show a less-seen side of herself to only Reina and at times, Shuuichi, one that is more fiery, stroppy, and determined, unlike the blandly polite facade she presents to acquaintances at school.
Not much is known about Asuka and Kumiko's relationship, but it seems like they are on good terms. Asuka sometimes calls Kumiko as Kumiko-chan just as how Kumiko calls her as Asuka-senpai. However, Kumiko knows that Asuka hides something behind that energetic face, which becomes part of several reasons (such as both playing the same instrument) of which Kumiko hates Asuka's false shallow personality, particularly the part that Kumiko feels as treating [Asuka's] own friends like nothing important.
Toward the end, Kumiko sees Asuka off on the latter's graduation day and receives her music notebook as a gift. In this, Kumiko takes the opportunity to tell Asuka that she loves her, being a sister figure for a majority of the series with parallels to Kumiko's biological sister Mamiko.
Shuuichi and Kumiko are close childhood friends. However, in junior high, when Kumiko invites Shuuichi to her home to play with him, he calls her ugly and tells her to never talk to him again. However, when they step into high school, they start being on good terms again and can freely talk with each other about various topics. They get along so well that they do not talk to each other using the polite form of speech, except for in public places. It is later revealed that Shuuichi has a crush on Kumiko when he asks her to attend the Agata Matsuri with him. Kumiko seems unimpressed by his offer, and ends up accidentally asking Reina. Shuuichi and Kumiko become very distant after these events, and have not since had a meaningful conversation, except for the one before their competition performance, when Shuuichi kept telling Kumiko it was going to be alright after which they had a fist bump. However, they still see each other as musical peers, and have been using each other for inspiration to improve their playing. Prior to their being distant, she seems to show a less-shown side of herself to only Reina and at times, Shuuichi.
In the light novel, Shuuichi tries to grow closer to Kumiko and nearly confesses to her the night before their final concert. He also gives Kumiko a hair decoration that’s shaped like the white flower Taki gave his wife before he proposed. Asuka mentions the hair decoration on Kumiko after her graduation ceremony and asks if it’s from her “boyfriend.” Kumiko stumbles and eventually says yes revealing that she is also in love with Shuuichi, and they start going out later in the light novels. In the anime, the plot subverts this in season 2 in which they don't start dating at all.
Hazuki is one Kumiko's first friends upon entering Kitauji High School, and Hazuki called Kumiko by her first name, which quite shocked Kumiko. Normally, Kumiko doesn't get along with sports-oriented people, but she gets along very well with Hazuki. They are on good terms and also, can talk with each other about various topics and so on.
When Kumiko found about that Hazuki liked Shuuichi, she tried to support them as much as possible, and she even declined Shuuichi's invitation to a festival, so Hazuki could go with him instead. This proves that Kumiko cares for Hazuki alot, and they are close friends.
Sapphire is also one of Kumiko's first friends upon entering Kitauji High School. Hazuki, Kumiko and Sapphire are almost always seen with each other. Kumiko and Sapphire also get along and they are on very good, friendly and close terms and support each other. Her friends and other people close to her call her, Midori.
Natsuki and Kumiko have a good and friendly relationship, as they both play the same instrument. However, it was later revealed that Natsuki was jealous of Kumiko since she played the euphonium better than her. Kumiko didn't realize this before she saw Natsuki practice very hard and when she didn't pass the audition. However, Natsuki later invited Kumiko to talk with her after the audition announcement. They talked about that and after that, they grew closer to each other.
At first, Kumiko thinks that Taki was very strict about practicing and going to the nationals, but she realizes that it is all to help the band. She respects him and sees him as a good teacher.
When Taki cuts Kumiko from an euphonium part in Crescent Moon Dance, he does it as respectfully as he can, without mentioning her name. Later, he offers a second chance to be given to her for the second round of competition, and reminds her to practice hard.
Taki takes notice when Kumiko begins to practice harder, and, like he does with other students, encourages her. In the second light novel, Taki buys Kumiko and Mizore Yoroizuka each a bottle of sweet green tea when he finds them practicing in the school building late in the evening.
When Kumiko is caught out in a typhoon with a broken umbrella, Taki drives her back to her apartment, and then gives her his own black umbrella before dropping her off.
After Taki fails to reach his own goals of having the band secure a gold award at nationals, he comes close to a breakdown in his office, shaking all over as he speaks, and tells Kumiko the full story of his wife, and her last living moments, and the promise he made to her. Kumiko consoles him with a few words. Taki thanks her, seeming to feel slightly better.
Kumiko, like most of the other students, thinks that Michie is a very strict teacher. However, she also respects her and calls her ''Michie-sensei'' even when not speaking to her directly. Michie also seems to know Mamiko and gets along with Akiko, Kumiko's mother.
Hashimoto is friendly to Kumiko, though she is not one of his percussion students. During a firework sparkler party, he sits down next to her in a quiet spot and, concerned, asks about Kousaka Reina's relationship with Noboru Taki. He accidentally lets slip during this conversation that Taki is a widower, and ends up telling Kumiko most of the full story.
Not so much known about their relationship, but Azusa and Kumiko where friends in junior high, and they later reunited when they were both performing at SunFes.