In most DBZ fanfiction, time travel often works through some sort of magical or technological device. For example, in some stories, Bulma creates a time machine similar to the one in the movie 'Back to the Future'. This allows characters to travel to different points in time within the Dragon Ball Z timeline.
One great 'Time Travel DBZ Fanfiction' is 'Back to the Future Saiyan Style'. It has an interesting plot where Goku travels back in time to prevent a major villain from ever rising to power. The character development is really well - done, and it stays true to the spirit of the Dragon Ball Z universe.
Vegeta is also quite popular. His pride and his constant strive for power can be really interesting when combined with time travel. He might go back in time to prove himself even stronger than he was originally or to correct past mistakes that made him look weak.
One interesting concept is how time travel affects character development. For example, a character might go back in time and change a key event, which then alters their future self in unexpected ways. Another concept could be the exploration of alternate timelines. Maybe a character's time travel creates a whole new reality where different battles play out and relationships shift.
The 'dbz kakarot time machine sub story' is likely about the time - traveling aspects in the Dragon Ball Z: Kakarot game. It might involve the use of the time machine to change events, meet different versions of characters, or unlock hidden storylines.
I'm not entirely sure, but it might work by setting specific time coordinates. Just like in the Dragon Ball Z series where they set the time and destination to travel to.
Goku might respond with his usual naivete and excitement. He'd probably be entertained by the new and wacky versions of himself and his friends in the fanfiction. Vegeta, on the other hand, would likely be furious at any portrayal that made him seem less than the proud Saiyan prince he is. Piccolo might be more indifferent, but still a bit curious about how the fanfic writers came up with some of their ideas.
One interesting plotline could be Gohan traveling back in time to prevent certain tragedies. For example, he might go back to stop the death of his father, Goku. This would change the entire course of the Dragon Ball Z story. Another could be Gohan using his knowledge from the future to train his younger self more effectively, leading to a much stronger Gohan at an earlier stage.