It might develop with Naruto being initially confused and a bit angry at being taken away. But as Minato starts to show him new techniques or tell him stories about his past, Naruto would gradually come to understand. For example, Minato could show Naruto how to better control his chakra, which would be a major development in the fanfiction. And as they spend more time together, their father - son relationship would strengthen, which would be a central part of the story's development.
One possibility is that Minato takes Naruto away to a secret training ground. There, he could start to teach Naruto advanced jutsu that he never got to pass on before. This would be a way for Minato to make up for lost time and help Naruto become even stronger.
It would be very close. Naruto has always longed to know his father, so when he travels back in time to meet Minato, they would quickly bond. They'd probably start training together right away.
At first, Minato might be a bit confused and in awe of female Naruto. He would see her as this strong and powerful figure who has taken him in. As time goes on, he would start to trust her more and look up to her as a role model. Naruto, on the other hand, would be very protective of Minato and would want to ensure he grows up to be a great ninja.
If Naruto takes Nagato's eyes in fanfiction, the development could be quite dramatic. Naruto would need to train to master the powers of the eyes. This could lead to him going on a journey to secluded places to practice without endangering others. His friends might try to help him, but they may also be in over their heads. Enemies could band together to try and take the eyes from Naruto, seeing it as an opportunity to gain ultimate power for themselves. This would create a lot of action - packed and tense situations in the story.
The story could develop with Obito training Naruto in his own way, away from the values of the Hidden Leaf. Naruto might grow up with a different set of skills and a different view of the ninja world. There could be a lot of internal conflict in Naruto as he discovers the truth about his origins and the fact that he was taken from his village. This could lead to a showdown between Naruto and the Hidden Leaf at some point, or Naruto might try to find a middle ground between Obito's teachings and his own sense of right and wrong.
Their relationship would likely develop with a great deal of mutual respect. Naruto has always looked up to Minato, and now that he's the 7th Hokage, Minato would see Naruto as a worthy successor. They would start to understand each other on a deeper level, sharing their dreams and fears.
Naruto's personality would be different. He'd be less lonely - driven, as he'd have Minato's love. He might be more focused on being a great ninja like his father from the start.
Naruto's personality would be a blend of his own natural exuberance and the more composed natures of Minato and Kushina. He might be less impulsive as Minato's strategic thinking would influence him. Also, Kushina's strong - willed nature would make him more determined and less likely to be swayed by others' negative opinions.
Naruto's character would develop in a more balanced way. Minato was calm and intelligent. So Naruto would learn those traits from him. He wouldn't have the same insecurities as in the original story. He might be more of a natural leader from the start, as Minato was also a respected leader. And he would have a better understanding of the village's politics and history because Minato would be there to educate him.
During their journey of running away, they would have a lot of time to talk and understand each other's values. Gaara, who has always been a bit of a loner, would start to appreciate Naruto's company. Naruto would be constantly there to cheer Gaara up when he has his dark moments. Their relationship would become one of mutual support, where they can be completely honest with each other and grow both as individuals and as friends.