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 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.
Well, he could encounter new allies and enemies in different villages and face all kinds of challenges on his own journey.
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.
Well, it would be a big loss for Konoha. Naruto is a powerful ninja, and his absence would mean one less strong fighter to defend the village. Ino, too, has her unique skills in mind - related jutsu and communication. Their departure might also lower the morale of other ninja, especially those who were friends with them or looked up to them.
One reason could be a special mission. For example, there might be a threat that only Naruto can help with in Suna. Maybe there's a new type of enemy that has some connection to Suna's unique chakra sources and Naruto's abilities are needed to figure it out.
One possible reason could be false accusations against him. Maybe he's wrongly blamed for a crime he didn't commit, like being framed for a major attack in the village. This would lead the village leaders to force him out to maintain 'peace' and security.