Well, perhaps they leave to find answers about the Uchiha clan. Naruto, being Sasuke's friend, decides to accompany him. They could end up in some ancient ruins where secrets about the Uchiha are hidden. Along the way, they have to face challenges like lack of supplies and hostile ninjas who don't want them to uncover those secrets.
In this kind of fanfiction, one possibility is that they go on a journey to discover new jutsu together. They might leave the Hidden Leaf Village to find a place where they can train without the constraints of the village rules. Maybe they encounter new and powerful enemies that force them to rely on each other more than ever, strengthening their bond.
They might leave because they want to start a new life away from the chaos and politics of the village. Naruto and Sasuke could head to a remote island. There, they build a small settlement and start training the local kids in ninjutsu. They also use this time to reflect on their past and heal from all the traumas they've been through.
If you want the clearest chapter reference, go to Chapter 8. Early on, The resentment that had barely calmed among the crowd surged back two...; later, the chapter leans on Some of them were relatives of the Jiang Family, who felt it improper to....
Chapter 841, "Isn't this Daniu?", is the clearest early answer because it turns Daniu's situation into a concrete plot event. The chapter turns that into something concrete by moving from Zhao Lang took the photograph and glanced at it, chuckling, "It...
Chapter 912, "Identical Four Pillars of Destiny?!", is the clearest early answer because it turns Lin Manru's situation into a concrete plot event. A useful checkpoint is the way the scene starts with But the repeated blows made Wan Huixia's husband lose his confidence...
If you want the clearest chapter reference, go to Chapter 20. Early on, Several people turned their heads upon hearing the sound and saw three...; later, the chapter leans on She was clad in a long black dress, and the dull color did not hide her....
If you want the clearest chapter reference, go to Chapter 49. Early on, My phone!!!" Jiang Yan's roar echoed continuously through the mountains in...; later, the chapter leans on Watching helplessly as his latest-model phone was taken away by a....
If you want the clearest chapter reference, go to Chapter 67. Early on, I'm pissed," Jiang Yan cursed again, "These people are too much, aren't...; later, the chapter leans on They expect us to fend for ourselves even to sleep and to eat—I mean, at....
A practical answer is Chapter 51, because it turns the idea into a concrete scene. The chapter opens with Oh, okay." After uttering two syllables indifferently, Jiang Li bent down..., adds pressure through With this in mind, she looked up and scrutinized the worried faces in..., and ends on If I.
The best chapter pointer here is Chapter 40, "The National Museum Speaks!". It opens with [Jiang Li claims she copied this herself? and closes on The handsome guys on this show are so difficult to deal with..