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 power and abilities, and has said before himself that he doesn't think he will ever reach Saitama's level of 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 Saitama wields, 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 came out victorious without a scratch on his body.
Genos has a strained relationship with Tatsumaki due to her smashing him through a wall several times or sending him flying away.
The relationship is not helped by her rude treatment on Saitama, which infuriates Genos a lot, and often tries to defend his sensei from her.
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 truly is.
King seems to be somewhat worried about Genos's tendency to use Saitama as a reference to get stronger.
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 with Saitama, Bang often attempts to be a master or mentor for him, but Genos rarely listens to his advice.
Like Tatsumaki, his relationship with Fubuki is rather strained, while arguably not as bad as the former. Genos still holds Fubuki in some contempt as noted when he congratulated his mentor on nearing B-Class but voiced his misgivings about said particular class's highest ranked heroine. 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 berates Fubuki's attempts to have him join her group. Tensions of which are said to insinuate from her attempted targeting of Saitama for rookie crushing.
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 father figure. Kuseno in turn cares about Genos and helps him anyway he can. Genos discusses with him his opinion on Saitama in admiration, something Dr. Kuseno has noted on several occasions.
Along with Saitama, he is the person Genos most loves and admires because of his intelligence and help, and has gone as far as standing up for Saitama because the latter called him "old man", though he later apologized for his attitude after Saitama told him to not take it so seriously.
Dr. Kuseno is also happy for the impact Saitama has had on Genos, as it helped the cyborg recompose his emotions and suppress his thirst for vengeance.
Like Saitama, Genos strikes an odd (even harsh) relationship with Charanko, mostly because he also refuses Bang's offer to join his dojo and learn martial arts. Genos is even more rude than his master in this regard, as he didn't hesitate to chock Charanko and beat him were he disrespect Saitama and him too much, he also doesn't show a lot of respect towards his abilities as well. However, they seem to be in more polite terms than before, but still they haven't built a friendly relationship like Saitama has (in his own way) done.
Due to Sonic's 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 (he actually wanted to avoid his house being destroyed).
Genos does not respect Garou, seeing him as just another thug at first, and is later surprised by his strength. He comes to recognize Garou is a real threat, and is impressed by Bang's training of him.