Perhaps Vegeta kidnaps Goku to get some secret training method from him. Vegeta is always eager to become stronger, and he might think Goku has some hidden knowledge that could help him surpass his limits. So he takes Goku to a secluded place to force the information out of him.
Well, one possibility is that Goku's friends would immediately start a search and rescue mission. Vegeta, being his rival but also an ally, would probably be at the forefront. He'd be furious that Broly dared to kidnap Goku. They'd likely track Broly's energy signature to wherever he's taken Goku. Another thing could be that Goku, while kidnapped, tries to reason with Broly. Maybe he discovers that Broly has some deeper issues or motives for the kidnapping that could potentially be resolved through conversation.
In the 'goku meets king vegeta fanfiction', there could be a power struggle at first. King Vegeta is proud and Goku is powerful. However, as the story progresses, they could come to respect each other's strength. Goku might show King Vegeta new ways of looking at power, not just as a means of domination but also for protection.
Well, they might start off as rivals. Vegeta could be initially dismissive of young Goku's power. But as the story progresses, they could end up teaming up against a common enemy, like Frieza perhaps. And along the way, Vegeta could learn some valuable lessons about humility from Goku.
In a 'Vegeta meets Chibi Goku' fanfiction, it could be really fun. Chibi Goku might be curious about Vegeta's Saiyan pride and constantly ask him questions. Vegeta would probably get annoyed at first but gradually warm up to the little guy. They could also have some training sessions together, with Vegeta trying to teach Chibi Goku some of his more advanced techniques, and in the process, they become friends who respect each other's abilities.
One possibility is that Gohan's friends and family, especially Goku, would immediately start searching for him. They would use all their powers and resources to track Vegeta's whereabouts. Goku might be enraged and ready to fight Vegeta to get his son back.
One possible plot could be that Vegeta kidnaps Bulma because he believes she has some technology that can help him gain an edge over his enemies. He takes her to a secret location and demands she work on it for him. Bulma, being the feisty and intelligent woman she is, tries to outwit Vegeta at every turn while also gradually getting to know a more vulnerable side of him that he doesn't usually show.
In fanfiction, if Goku and Vegeta go back in time, they might change the course of history. For example, they could prevent some of the villains from ever becoming powerful. They may also meet their past selves, which could lead to some really interesting battles or training sessions. And they could introduce new techniques to their past counterparts, making the Saiyans even stronger earlier in the timeline.
Well, they could change the entire timeline. Maybe they'd go back and team up with their younger selves to take on foes that were previously too powerful. They might also influence the relationships between characters. For instance, Vegeta could potentially have a different attitude towards the Z - fighters earlier, which would change the dynamics of their group. It could lead to a whole new set of adventures and battles that we haven't seen in the original story.
They might change the course of history and prevent some major disasters.
Perhaps Vegeta Jr could challenge Vegeta to a sparring match. Vegeta, being the proud warrior he is, would accept. During the fight, Vegeta Jr might display some new techniques he's developed, which could both surprise and intrigue Vegeta. And in the end, they could form a stronger bond as they realize they share the same values and goals as Saiyans.