This article contains major plot spoilers from the light novels that have not been revealed in the anime. Proceed at your own risk.
Kitauji High School
In a flashback to their time in middle school, the first episode shows Kumiko noticing Reina crying during the revelation of their middle school band's ranking-- a high placement that left the band slightly short of qualifying for the next round of advancement-- and believed her to be crying out of joy. However Reina was actually dismayed that she had lost her last chance to go to the nationals as a middle school student, she then ran away sobbing, leaving Kumiko speechless and guilty.
Due to this incident, the two enter high school with an uncertain relationship. Kumiko tries several times to confront Reina about what had happened, but fails every time due to her own awkwardness while Reina simply reacts as her usual self. Eventually, after a few tentative conversations, they gradually grow closer and develop a deep bond, apparent in their mutual blushing and occasional embarassment when they see each other.
In episode 8, Kumiko accidentally invites Reina to attend the Agata Festival with her. Still wary of each other, both girls reluctantly follow through with the unexpected chain of events that ensue. However, as they start talking about various topics, they eventually move to a first-name basis by the end of the festival.
With their friendship restored and stronger than before, Kumiko is probably the closest friend Reina has. They share a unique bond of trust, and have exchanged "confessions of love." Kumiko also states that she will always stay by Reina's side when she would compete against Kaori, claiming that she "wouldn't abandon [her], even if it costs [her] life."
Kaori is Reina's senior in the trumpet section. She is friendly to Reina because she is younger then her, and Reina returns her respectful attitude. They share a healthy and polite rivalry in the band.
Yuuko, a second-year trumpet player, harbors strong jealousy against Reina both because Reina outperforms the entire trumpet section and because Yuuko is jealous of Reina on behalf of Kaori, with whom she is obsessed. Yuuko sometimes expresses annoyance whenever Reina plays outside of rehearsal, but Reina still treats her with respect, knowing her own place as an underclassman.
Throughout the earlier episodes, there are hints that Reina has feelings for Taki, such as her unreasonable expression of frustration when Shuuichi mentions his disapproval of Taki-sensei to Kumiko. This is later confirmed when Reina confesses to Kumiko that she likes Taki in a romantic way.
In the light novels, Reina has a daily early-morning ritual of visiting Taki's office first thing when she arrives at school, before practicing trumpet and then heading to her first class. She also finds any excuse at all to visit Taki's office, no matter how pointless.
It is also revealed in the light novels that Taki's father, Tohru Taki, taught the band alongside Reina's father for many years at Kitauji. Taki's sister and Reina also knew each other a very long time, so Taki had known Reina since he was a teenager. The younger Taki used to cheer up Reina when she was a young child by playing music with her.
In the light novels, when Reina meets Satomi Niiyama and suspects her to be Taki's girlfriend (later proven false), she smashes a drinking glass on the ground in rage. She begins to perform her solo during rehearsals with noticeably less emotion and sensitivity, causing Taki to grow concerned as the competition approaches. However, in the anime, Reina does neither of these.
At the end of the third novel, Reina confesses to Taki that she loves him after the final concert while all the bands/mentors are being honored. However, because of her choice to speak to him in this situation, Taki takes her words as a sign of respect from a student rather than of romantic interest. Despite knowing that her intentions were misunderstood, Reina smiles anyway, as observed by Kumiko, and seems content with his polite response.
Reina later learns from Taki personally about his wife and her passing.