Maybe they leave to find a cure for a strange disease that has started to affect the ninja world. Jiraiya has some clues about a special herb in a far - off region and they embark on a journey. During this, Naruto learns about the importance of perseverance from Jiraiya. They face many enemies who don't want them to succeed in their mission.
In some fanfictions, they might leave to train Naruto in isolation from the distractions of the village. Jiraiya takes Naruto to a secluded island where they can focus solely on Naruto's growth. There, Naruto not only hones his combat skills but also his understanding of the ninja way. Jiraiya shares his own life experiences and wisdom, which has a profound impact on Naruto's development.
Well, in this kind of fanfiction, there could be a plot where Naruto discovers some secret teachings that Jiraiya had left specifically for him as his son. This could give him an edge in battles and also in understanding the politics of the ninja world. Maybe Naruto would use these teachings to bring about a new era of peace in the shinobi world, with a different approach compared to what we saw in the original series.
One possible reason could be that Naruto feels he has learned all he can from Jiraiya at that point. Maybe he has reached a level in his training where he believes he needs new experiences and challenges that Jiraiya can no longer provide.
One possible plot could be that after Naruto leaves the village for a long mission, Ino realizes she has feelings for him. When he returns, she starts to subtly show her affection. For example, she might always be around to offer him medical help after his battles, using it as an excuse to be close to him.
Some fanfictions might feature rogue ninja as antagonists. These rogue ninja are after the bounty on Naruto's head or they are trying to steal Jiraiya's knowledge. They attack Naruto and Jiraiya during their travels, making it difficult for them to reach their destination.
One interesting Naruto and Jiraiya fanfiction could be about Jiraiya teaching Naruto some super - secret and powerful jutsu that he never got to in the original story. Maybe it's a jutsu that combines elements in a unique way.
One popular plot is Jiraiya going back in time and using his knowledge to prevent the Uchiha massacre. He could train Sasuke differently, or even find a way to resolve the conflict between the Uchiha and the village. Another plot might be Naruto traveling back with Jiraiya. They could team up to change events that led to the rise of Akatsuki, like stopping Nagato from going down the dark path. There are also plots where Jiraiya tries to change his own past mistakes, such as not being able to save Orochimaru from the darkness earlier.
Perhaps a plot where Naruto, as Jiraiya's apprentice, goes on a secret mission with him to spy on a hidden enemy organization. During this mission, he has to use his wits and the skills Jiraiya taught him to stay alive and gather information.
A possible plotline is Naruto and Jiraiya going on a journey where they encounter a new and mysterious organization. This organization has a connection to Naruto's past, perhaps related to the Nine - Tails. Together, Naruto and Jiraiya have to figure out the motives of this group and stop them from causing chaos in the ninja world. It would be full of action, mystery, and character exploration.
A great plotline could be the family - like bond that forms between them. Jiraiya, being like a father figure, and Tsunade, like a mother figure, they could have Naruto experience a more stable family life. This could lead to Naruto having different emotional growth. He might become more confident not just in his ninja skills but also in his relationships. Also, they could help Naruto discover more about his parents through their own memories and knowledge.
In the fanfiction, there might be a moment where Jiraiya takes Naruto to a secret training ground. This place has some special chakra - enhancing properties. Naruto has to endure a series of tough physical and mental challenges there. He struggles at first but gradually overcomes them, growing stronger in the process.