One possible adventure is that they travel to different ninja villages in disguise. Naruto gets to learn about the cultures and ninja techniques of other villages. This helps him gain a broader perspective on the ninja world and also gives him ideas for new jutsu. Jiraiya might also introduce Naruto to some of his old friends or informants in these villages.
They might have adventures in training Naruto to control the Nine - Tails' chakra better. Jiraiya could take Naruto to different secluded places where he can practice without distractions. Maybe they encounter strange and powerful beasts along the way, which Naruto has to learn to fight using his new skills.
Well, he could encounter new allies and enemies in different villages and face all kinds of challenges on his own journey.
Naruto's character develops through the training he receives from Jiraiya. He becomes more disciplined. For example, he learns to wake up early every day for training without complaining like he used to in Konoha. His ninja skills also improve, which boosts his confidence.
One possibility could be that Naruto leaves Konoha and stumbles upon a village where everyone talks in rhymes. He tries to communicate but ends up making a comical mess of things. For example, when he asks for food, he says it all wrong and the villagers think he's performing a new kind of comedy act.
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.
The relationship between Konoha and the Uzumaki clan might be further strained. With Kushina leaving, Konoha would lose an important link to the Uzumaki. And other Uzumaki members might see this as a sign that Konoha is not a safe place for them.
Another reason could be that he witnesses corruption or unfairness within the village leadership. If he sees that the system is rigged against certain groups, including himself or his friends, he could grow to hate Konoha and decide to leave.
One possible reason could be to seek stronger opponents outside of Konoha. Naruto might feel that he has outgrown the training opportunities in the village and wants to face greater challenges.
One possible reason could be that Naruto feels he has learned all he can from Jiraiya at that point. Maybe he wants to explore his own path independently and test his skills without Jiraiya's constant guidance.
Konoha would initially be in shock. Naruto is a hero to many in the village, and his sudden departure with Itachi, who was once considered a traitor, would cause a lot of unrest. The villagers might start to question the leadership of the village, wondering why Naruto left and if there was something they could have done to prevent it.
Jiraiya's reaction might be complex. He might be in awe of Naruto's new power but also sad about his departure. He could see it as a natural progression in Naruto's journey, yet still feel a pang of loneliness. He might start to reflect on his own life and training methods, wondering if he could have done something differently to keep Naruto with him. At the same time, he would be eager to see what Naruto will achieve in the world with his newfound abilities.