Issei, at first, hated Azazel because he was a Fallen Angel, but learned later on that he is a good man. Azazel later became the Occult Research Club's advisor in Volume 4, and aided the Gremory team on how to better improve themselves as a team. He also helped out in progressing Issei and Kiba's Sacred Gears, as well as helping Issei learn how to achieve Balance Breaker and control the Trianna. He also shows a great deal of concern for Issei after his death but was relieved to hear his spirit is alive.
Akeno initially hated Azazel because Azazel is a Fallen Angel like her father. Azazel, on the other hand, cares a lot for Akeno because she is the daughter of his good friend Baraqiel, but also specifically because he feels guilty about her mother's death. Azazel believes that he was responsible for it since it was him who called Baraqiel out on a mission that day, thus leaving Akeno's mother undefended. Azazel treats Akeno like his own daughter and constantly asking Issei to protect Akeno.
One of Azazel's closest friends, Azazel cares deeply for Shemhazai while Shemhazai respects him.
One of Azazel's closest friends, Azazel cares a lot for Baraqiel and feels guilty for having Baraqiel help him which indirectly lead to Baraqiel's wife, Shuri Himejima, death and constantly tries to find a method to compensate Baraqiel.
Azazel has a respectful relationship with Sirzech since they were both representatives for the treaty in volume 4 since then they have gotten along even worked together to deal with situations like the Diodara incident in Volume 6 and calming Issei down to release him from Juggernaut Drive also in Volume 6.
Azazel has a deep rivalry with Michael even before he became a Fallen Angel. This is mainly because Michael once revealed Azazel's Artificial Sacred Gear's design throughout the Heaven and was made fun because of it.