In the Dragon Ball universe, there are many strange and wonderful things. King Kai might have access to some rare or magical items. He could use one of these items to help Goku's tail grow back. For example, there could be a special fruit on his planet that has regenerative properties. When Goku consumes it or King Kai uses it in some other way, it triggers the growth of the tail.
It could also be a symbol of a new power - up. In the Dragon Ball series, new physical changes often lead to increased power. So, getting his tail back might mean that Goku is about to reach a new level of strength. Maybe he'll be able to transform into a new form that is related to his Saiyan heritage and the presence of the tail. This could give him an edge in future battles against stronger opponents.
Well, he would probably be a very different person. He might be more in tune with the Saiyan warrior culture. He could be more aggressive and less empathetic compared to the Goku we know. He would grow up learning the ways of the Saiyans on Planet Vegeta, which means he'd be trained to fight and conquer from a very young age.
Adventure is another typical theme. They could go on adventures together, either in the Other World or on Earth. These adventures can range from fighting new villains to exploring uncharted territories. This theme allows for a lot of action and excitement in the fanfictions.
One could expect an exciting adventure. Maybe it'll involve Goku's power being used in a unique way to protect the timeline that Supreme Kai of Time oversees. It could also show a different side of their relationship, perhaps a friendship that develops as they face challenges together.
The plot might center around Goku being raised by Beerus in a more secluded place. Beerus could be training Goku to be his successor as a god of destruction. Goku would have to learn about the balance of destruction and creation, which is different from his usual Saiyan power - up goals. This would also mean that Goku would have a different view on the universe and its inhabitants. He might be more detached in some ways, but also more powerful and with a different moral code influenced by Beerus.
His power would develop more in line with Saiyan traditions. He'd probably have a stronger base power due to the intense training environment on Planet Vegeta. There would be more emphasis on raw power and combat skills from a young age.
His social skills might be different. Instead of the somewhat naive Goku we know, he could be more aware of social norms. Bulma could teach him about proper behavior, but at the same time, his Saiyan impulsiveness would still shine through at times. So he might be a strange mix of a well - mannered person in some situations and a wild fighter when provoked.
His character would develop a more strategic side. Beerus is a very strategic fighter, and Goku, being raised by him, would learn to think more before acting in battles. He would also become more aware of the power hierarchy in the universe at a younger age, which could make him more cautious or more ambitious, depending on how you look at it. Additionally, his Saiyan pride might be tempered by Beerus' teachings, making him more of a team player in some ways.
Another storyline could involve Supreme Kai of Time getting into trouble in a different timeline and Goku having to rescue her. Along the way, they discover new forms of power and new allies.