Well, it could be about Goku Jr learning some of Goku's most powerful moves. Maybe Goku teaches him the Kamehameha. They might also go on an adventure together, like fighting a new and powerful enemy that neither could face alone.
Another potential rival could be a young, prodigious martial artist from a new school of martial arts on Earth. This character could be jealous of Goku Jr's fame and constantly try to outdo him in battles and in the eyes of the public. He might use unorthodox techniques that Goku Jr has to quickly figure out how to counter.
Maybe an old enemy of Goku's who has somehow managed to return. This enemy has heard about Goku Jr and figures that by attacking him, he can get revenge on Goku. This could be someone like Frieza who has found a new way to be resurrected and is more powerful than ever.
To develop Goku Jr's character further, he could be given more complex emotions. For example, showing his fear in the face of truly overwhelming enemies, rather than always being fearless like Goku. This would make him more relatable. Also, exploring his relationships in more depth, like his friendship with Pan. Maybe they have a falling out over different views on how to protect their world, which forces Goku Jr. to grow and understand different perspectives.
Well, Vegeta would be a likely character. He might be initially suspicious of the female Goku. Maybe he thinks she's a threat or just can't believe there's another Goku - like being. He could be there to add some comical moments as he tries to understand the situation.