One challenge is getting the character of Vegeta right. He has a very distinct personality, with his pride and his desire for power. If you don't capture that essence, the story can feel off. For example, if you make him too meek or too friendly too quickly, it doesn't fit his character.
Another one is 'Vegeta in the Harry Potter World'. Here, Vegeta has to adapt to the magical environment. His initial confusion and then his determination to master the magic (in his own way) is really fun to read. The way he interacts with the Hogwarts students and teachers is full of comical and exciting moments.
Power is a common theme. Since Vegeta is extremely powerful, seeing how his power compares to characters from other universes is always interesting. Another theme is culture clash. For instance, when Vegeta enters the Harry Potter world, his Saiyan warrior culture clashes with the wizarding culture. Pride is also a theme. Vegeta's Saiyan pride often plays a role in how he interacts with characters from other fandoms.
In a strange turn of events, a portal opened up and Vegeta found himself in Gotham City, home of Batman. Vegeta was confused but ready for a fight. Batman, always cautious, observed Vegeta. Eventually, Batman convinced Vegeta that there was a greater threat that required their combined efforts. The Justice League, including Wonder Woman, Flash, and Aquaman, joined them. Vegeta was impressed by their diverse abilities. During the battle against a mind - controlling villain, Vegeta's brute strength and the Justice League's strategic planning worked in harmony. Vegeta learned about teamwork from the League, and they all respected his warrior spirit in the end.
For Goku's character development, you can explore his family side more. Maybe he realizes that his constant pursuit of strength is affecting his relationship with his family in the fanfic. Vegeta, on the other hand, could start to open up more about his past traumas. He could learn to let go of his pride a bit when he sees Goku's selfless attitude towards others. This would be a significant development for him.
The plot could start with Vegeta being exiled from his own planet due to some political intrigue. He ends up on Earth, where he crosses paths with the Justice League. At first, there is tension as the League is unsure of his intentions. But when a powerful enemy, like Darkseid, starts to target Earth, Vegeta decides to help. He teams up with Superman, and they have an epic battle of strength. Meanwhile, Batman uses his detective skills to figure out a weakness in the enemy's plan. Wonder Woman, Aquaman, and Flash also play their parts, with Vegeta adding his Saiyan speed and power to the mix. In the end, they defeat the enemy, and Vegeta decides to stay on Earth for a while to learn more about these strange heroes.
Well, I like 'Vegeta in the DC Universe' a lot. In this fanfiction, Vegeta's might is pitted against the likes of Superman and Batman. It's fascinating to read about the power dynamics. Then there's 'Vegeta and the LOTR (Lord of the Rings) Crossover'. Here, Vegeta's presence in Middle - earth changes the course of many events. Also, 'Vegeta in the Avatar World' is quite good, with his Saiyan abilities clashing and sometimes meshing with the bending arts.
Perhaps an ancient evil Saiyan who was banished long ago. This Saiyan would have powers that are unknown to Goku and Vegeta. He could be awakened from a deep slumber and is angry at the current state of the Saiyan race. He sees Goku and Vegeta as traitors for living among the Earthlings and decides to destroy them and take over the universe. His knowledge of ancient Saiyan techniques would give him an edge in battles against Goku and Vegeta.
One idea could be Vegeta arriving in the DC universe during a major crisis. He might initially be seen as a threat but then teams up with Superman. Their powers are quite different, with Vegeta's energy blasts and combat skills from Saiyan training, and Superman's super strength and flight. Another idea is Vegeta getting involved in the Gotham underworld, facing off against villains like the Joker. He could use his no - nonsense attitude to deal with the chaos there.
One interesting concept could be Vegeta's power levels in the crossover. He might gain new abilities or face opponents he wouldn't in the original series. For example, if it's a crossover with a magic - based universe, seeing how his Saiyan strength interacts with magic would be cool.