Saitama met Genos after the latter was defeated by a monster and tried to self-destruct, but was saved by Saitama. After being saved Genos asked to be Saitama's disciple, to which Saitama nonchalantly accepts, thinking he wasn't serious. Though Saitama was annoyed by the thought of having a student, he gradually warmed up to Genos, even becoming angry after hearing he was defeated by the Sea King and allowing him to practically live in his apartment, though he was irritated by it at first.
Genos has a great deal of respect for his master. He is in complete awe at Saitama's phenomenal physical prowess and abilities, and states himself he doesn't think he will ever reach Saitama's power. He is firmly dedicated to stay as a disciple to Saitama, and strongly rejects anyone who tries to make him their disciple or lackey, feeling the need to protect his master's honor. He greatly respects Saitama's humbleness and carefree attitude, as he doesn't report monsters being slain by Saitama, or tries to use his connections as an S-Class Hero to get Saitama privileges. He only ever once lashed out at him, when Saitama explained the origin of his unlimited power was simple training, since Genos was curious to find out and still is. He also knows when to let his master handle the situation as he didn't stop Saitama from giving the weaker heroes all credit for the defeat of the Deep Sea King, even though it could have strengthened the reputation of his master. Genos tries to do whatever he can for his master, and will express extreme disdain when he unable to perform the tasks asks for Saitama, no matter how insignificant or nominal they may be.
The running gag in their relationship is that Genos seems convinced that Saitama suffers under his lack of hair and tries to "help" him with that problem as often as possible (though it's somewhat understandable, since Saitama only ever gets agitated when someone mentions his baldness).
Despite his confidence and faith in Saitama's limitless strength, he cannot comprehend the true power of Saitama since he thinks Saitama will need helps to defeat some powerful monster that Genos himself cannot defeat. The prime example is Gouketsu. However, unknown to him, Saitama always come out victorious without any scratch from his body.
Genos has a strained relationship with Tatsumaki due to her smashing him through a wall several times or sending him flying away.
While the two don't interact much Genos possesses a certain amount of respect towards Bofoi due to his powerful weapons and scientific genus. According to Drive Knight, Bofoi is Genos' enemy, causing Genos to become interested in Metal Knight.
While the exact nature of their relationship is unknown, the two are shown to be friends. Like most characters Genos is unaware about how weak King is.
The two get along fairly well. Bang passes advice onto Genos whenever he needs it and Genos holds a certain amount of respect towards Bang due to his martial arts skills and rank in the hero association. Like with Saitama, Bang tried to recruit Genos for his dojo but is turned down, and this constant attempt of recruitment has caused Genos to lash out at him on occasion.
Like Tatsumaki, his relationship with Fubuki is rather strained, although arguably not as bad as Tatsumaki. The two don't often communicate with each other, but they don't appear to have a friendly relationship. This is seen when Genos rejects and slightly berates Fubuki's attempts to have him join her group.
The two only briefly interacted with one another. All the information Genos knows about Drive Knight is from the Hero Association and has never met Drive Knight prior to their interaction. How much Drive Knight knows about Genos is unknown. However, he was willing to warn Genos about Metal Knight. This suggests Drive Knight has some knowledge of Genos beyond the Hero Association and is willing to trust him. So far, Genos appears to be the only Hero who is warned about Metal Knight.
Genos is greatly indebted to Kuseno for saving his life and considers him a friend. Kuseno in turn cares about Genos and helps him anyway he can. Genos will discuss with him his opinion on Saitama in admiration, something Dr. Kuseno has noted on several occasions.
Due to his obsession with Saitama, Genos despises Speed-o'-Sound Sonic. He also considers Sonic a rival and a reason to become stronger. He offers Saitama to eliminate him for his master, but Saitama refuses to do so, saying Genos wasn't much better ability-wise.
Genos has shown immense respect for the monster martial artist. Believing it would take the combined might of Saitama and the S-Class heroes to defeat him.