Obviously, Goku would be a main character he'd interact with. If he goes back to when Goku was a kid, he could train with him, or try to convince Goku's grandpa to let him train Goku in a different way. Another character would be Bulma. If he goes back far enough, he could meet a younger Bulma and perhaps change their relationship dynamics. He might also interact with his younger self. This could lead to some really interesting self - reflection and growth as he tries to guide his younger, more hot - headed self.
In some 'Vegeta goes back in time' fanfictions, Vegeta might go back to the time when the Saiyans were still under the rule of King Vegeta. He could try to change the Saiyan culture from being so war - like and brutal. He might introduce new ways of thinking and training that would not only make the Saiyans stronger but also more united. This could lead to a completely different future for the Saiyans, where they don't get wiped out by Frieza and instead become a more dominant force in the universe. He could also form different alliances with other races much earlier, which would change the power dynamics in the galaxy.
In the fanfiction, perhaps it focuses on the initial spark between them. It might show how Vegeta, with his tough exterior, starts to soften a bit when he's around Bulma. For example, he could start to notice small things about her, like the way she furrows her brow when she's thinking hard about something in her lab, and that starts to draw him to her.
In the original, their relationship develops gradually over time. But in the fanfiction, it could speed things up. For instance, they might have a more immediate connection, like falling in love at first sight instead of the slow - burn we see in the official story.
He could also become more strategic. Time travel would expose him to different situations and enemies. He'd need to plan better and think ahead. Maybe he travels to a time when the Z - fighters are all defeated, and he has to figure out how to prevent it. This would make him less impulsive and more of a long - term thinker.
At first, it might be a relationship based on mutual need. Vegeta needs Bulma's resources and intelligence to get back on his feet, and Bulma is intrigued by this alien. As time passes, they start to understand each other better. For example, Vegeta may start to open up about his past and his Saiyan pride, and Bulma will share her dreams and ambitions with him.
Who goes first really depends on the story's setup. It could be a character introduced early to establish the setting, or it could be a mysterious figure whose actions trigger the entire narrative.
You can expect her to be a bit nervous. It's normal. She might be overly cautious about her appearance and what she says. Just be there to support her. Give her some encouraging words.