They might encounter new allies. These could be people with different abilities and backgrounds. For example, they could meet a group of rogue ninjas who turn out to be good-hearted and help them on their journey.
The story could focus on their self - discovery. Away from Konoha, they have to rely on themselves more. Naruto might learn more about his family's past, and his friends could discover new aspects of their own identities and powers.
Sakura might be initially shocked and heartbroken. She has been Naruto's friend for a long time, and his sudden departure would leave a big hole in her life. She might try to convince him to stay at first.
As he is forced to leave, Naruto may become more cautious. In Konoha, he had some level of protection, but now on his own, he has to be more aware of his surroundings. This could also make him more empathetic towards outcasts, as he is now in a similar situation. His ninja skills might also improve faster as he has to rely solely on himself in battles and difficult situations.
His friends might feel betrayed. They would wonder how money could be more important than their friendship. Some might try to convince him to stay, while others could be angry and cut ties with him.
One reason could be to seek greater power. In fanfiction, the world outside Konoha might be seen as having more opportunities for growth. They might want to train with different masters or learn unique jutsu not available in Konoha.
The development varies a great deal. In some fanfictions, after releasing the Kyuubi on Konoha, Naruto is exiled. He then has to find his own path outside the village. Along the way, he meets new allies who help him understand his power better. He might also encounter new enemies who are interested in the Kyuubi's power. Eventually, he could return to Konoha as a more powerful and wiser shinobi to make amends.
Naruto's character can develop in several ways. Away from Konoha, he is forced to rely more on his own instincts and less on the support system he had in the village. He might become more self - reliant and learn to make decisions without the immediate guidance of his friends or mentors like Kakashi. For example, when facing a new type of enemy, he has to quickly figure out a strategy on his own. This makes him grow in confidence and his ninja skills improve as he has to adapt to different fighting styles and terrains.
Well, it could be that Sasuke decides to join Naruto in leaving. They've always had a complex relationship. Maybe Sasuke sees Naruto's reasons for leaving as valid and decides to support him, which strengthens their bond.
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, 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.
Often, his friends in Konoha will start to search for him. They might split into teams and scour the surrounding areas. For example, Sasuke could use his Sharingan to look for any signs of Naruto's departure. Meanwhile, back in Konoha, there could be unrest as the villagers start to worry about the safety of the village without Naruto.