This section is rich of spoilers until chapter 23. Be sure to want to know everything about Rin's relations before reading!!
My apologies for my bad English...
Rin & Yukio
Yukio is the most important person for Rin. He's his twin brother and the one that Rin considers as a goal to reach. Rin's biggest desire is becoming an exorcist, but this decision borns from the fact that Rin doesn't want to see again a person dying in front of his eyes. He considers himself culprit of the death of Father Fujimoto, his beloved father, and because of this, he decided to fight his true nature of Satan's son and become an exorcist able to protect who is important for him.
Rin doesn't know that Yukio's actions and desire is the same of him: to protect and to save his beloved nii-san.
Since they're totally opposite, the major part of the times they fight and have discussions for many reason, and even if Rin is the older brother (because he born two hours before Yukio), he seems to be the younger one.
Rin & Shiemi
Shiemi is the first person that Rin meets thinking that she remembers him himself.
She seems to be quiet and shy, but her desire is becoming stringer in order to protect Rin and Yukio, and obviously, she doesn't want to be useless. The meeting between her and Rin is important for both of them, because Shiemi finally leave the garden of her grand-mother and decide to go to school in order to become an exorcist.
In fact, Shiemi's reason was just to become stronger and to be useful to the others, but day by day she understands that she wants to become something more, she wants to be someone able to help Rin and Yukio.
Rin has a crush on her but in a first time she seems to be more interested in Yukio. After a lot of moments spent with Rin and trying to be more sincere with herself, maybe we can think that now she's in love with him.
Rin uses their mission to the Luna Park to ask her a date, and she accepts, also, she always search for him and it's obvious that she feels something special for Rin.
Shiemi is so grateful to Rin because is thanks to him if she found the peace with her mother, saying that she loves her. And even if in a first moment she felt feared by his demon nature, after izumo's words she understands that Rin is a good guy and she wants to be his friend and a support for him with all the heart.
Rin & Mephistopheles
Their first meeting was on the occasion of the funeral of Father Fujimoto. Since Rin found the truth about himself, realizing that he's a demon son of Satan, Mephistopheles reached the cemetery to kill him.
Rin decides to become an exorcist to kill Satana, and Mephistopheles decides to use him like a weapon against Satana. They don't have a deep relation and Mephistopheles is not interested in Rin's life (definitely, he's not like Fujimoto, that considered Rin as his real son and died in order to protect him), his only goal is using Rin, he's not important more than any ather weapon. Since Rin need to study, Mephisto usually provides to his upkeep giving him 2000yen per month.
Rin & Amaimon
Amaimon could be considered "Rin's uncle", but his interest for him is not a positive thing.
Amaimon wants to fight Rin and Satan's flames, he demands to costrict him to fight until the end in order to prove his power.
And he doesn't know good manners, he controlled and used Shiemi against Rin two times (to the Luna Park and in Kyoto), because he understood that she's important for him.
in order to pretect, Rin
decides to use his power and the blue flames, and finally Amaimon is happy, because he obtains the what he wanted.
Rin & Fujimoto
Fujimoto Shirou is the priest (and the exorcist) that took care of Rinand Yukio since their born.
He was the greatest and the most splendid father that Rin would desire, and only after his death, Rin understands how important was that man for him.
Fujimoto always tried to teach to Rin the meaning of his strenght, and how important is control his emotions and the latent power that he hides in himself.
Also, Fujimoto's desire was Rin's happiness, and this is the reason why his words for him weren't that kind, in moment of Rin's sadness. Because he needed to make him able to understand that one day, because of his power, surely he should remaine alone.
The memories of Fujimoto and him in childhool are Rin's happiest moments, and he always think about them considering his father's words as the most precious treasure he left to him, together with Kurikara, the sword ables to hide Rin's demon power.
Rin & Shura
Shura is the person
appointment to teach to Rin how he needs to use his sword, because she was father Fujimoto's disciple.
In a first moment she thought that Rin was a coward, because he didn't use his power for a reason, but knowing him better, she understood that Rin's reason is that we doesn't want to appear like a monster, and he tries to using his flames less than he can.
But Shura understands that they're the only way he has to fight, and he needs to control them, and obviously, Kurikara.
So, she accepts to become his mentor and she starts to teach him real lessons with realism but taking care of him like only a real older sister could do.
Rin & Suguro
The relation between Rin and Suguro has a sweet development, because in a first moment they considered the other one as a stupid and useless person.
Rin understood that Suguro is smart and so determinated to become an exorcist, and Suguro found in him a great fighter and a person that is always ready to help who is in difficult.
They're goal is to defeat Satan, and because of many reasons, they find a lot of similar sides into the other one (Suguro gifts a hairpin to Rin in order to fix his hair when he needs to study, and this is a sweet thought because he uses the same hairpin), and day by day they learn to be friends.
It's thanks to Rin if Suguro becomes able to understand his father and to respect him.
Rin & Izumo
I think that the relation bewteen Rin and Izumo is adorable. He's a hot-headed boy and she's tsundere, but they share special moments that really makes me think that they could be cute as couple.
Rin is the only person ables to see the deepest side of Izumo's heart. He's the one that gave to her his jacket to help, the one that was able to wipe aways her tears. And she's the only person that even finding Rin's true form and nature, never left his side. I really admired her in the moment in which she sits downs beside him into the plane for Tokyo, because no one wanted to do it because of fear. But Izumo isn't just an intuitive person, she's smart and alert, and she understood that Shiemi has feelings for him (but I don't think she understood that Rin has a real crush on her).
Rin & Kuro
Rin is the only one that was able to discuss and to help Kuro after Fujimoto's death.
Kuro was his familiar, and Fujimoto saved it in past, so Kuro was totally devoted to him. Rin was able to say the truth to him, and when Kuro found that Fujimoto is dead, finally it can find his peace, stopping to wait for him.
Rin decides to keep Kuro with him and they become friends. Since Rin is a spontaneous person, Kuro and him always show their love for the other one, especially Kuro, that loves to stay on a Rin's shoulder.