Giovanni is a mysterious character who rarely interacts with other characters aside from his team of Rocket grunts, and maybe a few others. Here I attempt to examine the various relations he has with particular characters, over the course of several mediums.
For the sake of this site I never refer to Jessie, James, and Meowth as Team Rocket. Team Rocket is actually a huge organization rather than a group of three comic relief characters. These three usually refer to Giovanni simply as "The Boss" and in actuality, they have very little interaction with him. Save for the one or two encounters they had through the animated series. In the episode "Battle of the Badge", they manage to bring him Misty's Togepi. When confessing that they have no idea what it does, he proceeds to scold them and yet make them take his place as the gym leader for the episode.
Giovanni's relationship with these three is rather peculiar. On one hand he obviously cannot stand them. They are pretty much worthless to him but are dedicated members of Team Rocket. (Though, in the first episode, it eludes to the fact that these three were actually at one point hardened criminals who were actually good at their job.) This may be the one reason Giovanni keeps them on the team. He does continue to supply them with seemingly endless funds (illustrating his great wealth) for unknown reasons. It could be gathered, however, that he does so to try and keep them out of the way of his bigger schemes.
Domino was a Rocket Grunt character who was introduced (and appeared solely) in Mewtwo Returns, which was a special long episode in Japan and was distributed right to DVD in North America. Domino was a competitent female character who was ever loyal and trusted by Giovanni. In almost the entire story she is constantly by his side, which greatly irritated Jessie, James, and Meowth.
Domino is proof in the anime that Giovanni can have a great relationship with another human being. At times he really regards her highly and trusts her, while she can invoke his anger, it's usually not at her but at the fact that his current plan is failing. Though their relationship is brief (only about 90 minutes long) it is a great illustration for Giovanni's personality.
The hero and his rival, Gary (Shigeru) and Ash (Satoshi), meet Giovanni ever so briefly. Ash meets him in Mewtwo Returns where Gary meets him in Battle of the Badge. Giovanni's relation with these two is the same as young trainer: They are pests. They meddle in his plans and try to win his Earth Badge. Though, when defeated, he will forfeit the badge honorably, but he aims to defeat. For Giovanni, his pride gets in the way before forming bonds with young trainers, much unlike Gym Leaders Misty and Brock who help Ash on his adventure.
In the game, Giovanni is frustrated and angered with the hero character who foiled his many plans. I would say that their relationship is existant but only in the case that they dislike each other. The hero sees Giovanni as a villain (as he is) and therefore cannot reciprocate a positive relationship.
In the manga, along with Sabrina and Lt. Surge, Koga is one of Giovanni's agents. These three are used to help Giovanni with his various schemes, which include following the hero (Red) in the guise of a researcher. Giovanni in the manga shows off his ruthless side when dealing with the Gym Leaders. They are more powerful than the Rocket Grunts and are also Gym Leaders, which can prove useful to Giovanni as he can exploit their powers.