They might encounter various new adventures. For example, they could come across a hidden village with unique ninja techniques. They may also face powerful enemies that are unknown to Konoha. Maybe they'll find a mysterious ancient artifact that holds great power and has to protect it from those who would misuse it.
They might meet a group of rogue ninjas who are actually good - hearted but have been misjudged. Naruto would be quick to try and befriend them, and together they could form an alliance to fight against a common enemy that they discover on their journey outside of Konoha.
Maybe they leave to find a better way to protect Konoha. They might think that by exploring the outside world, they can gain new knowledge and powers that will be useful for their village in the long run.
Perhaps they are leaving together to search for a cure for some new and dangerous disease that has started to affect ninjas. They believe that by traveling the world, they will find rare herbs or meet wise sages who can help them develop a remedy. Naruto's determination and Hinata's knowledge of medicinal plants could be a great combination for this noble quest.
Another reason might be that Naruto is seeking a greater purpose. In the fanfiction, his 'godlike' state could make him realize that there are threats out there that he can't face while being tied to Konoha. He may leave to protect the world on a larger scale, believing that he can be more effective away from the village.
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.
They could go to the Land of Fire's outskirts, where there are less - known ninja clans. Naruto might hope to learn from them and Anko could be interested in the unique fighting styles these clans possess.
Kiba would probably be shocked. He's Naruto's friend and also has a connection to Hinata. He might try to convince them to stay, saying things like Konoha needs them. But once he realizes they are determined, he would support them in his own way, like promising to visit them. As for Lee, he would see it as a very brave decision. He would be inspired by their courage to step out of their comfort zone and start a new life elsewhere.
Well, they could have adventures that are more personal in nature. Naruto and Shikamaru as brothers might have to deal with family - related issues within the village. Maybe there are some villagers who don't accept their 'brotherhood' because Naruto is an orphan and Shikamaru comes from a well - known family. They would have to prove that their bond is as strong as any blood - related brothers, through acts of kindness and heroism towards the village.
One reason could be to seek greater power. Maybe they feel that Konoha no longer offers the training or resources they need to reach their full potential. So they decide to leave and explore other places where they can become stronger.