Maybe Sasuke teaches him the Chidori in a moment of camaraderie. And for the Rasengan, it could be an older and wiser Naruto from the future who travels back in time to teach his younger self.
In some fanfictions, Naruto might learn Rasengan through the guidance of an alternative mentor. Maybe a powerful shinobi from another village takes him under their wing and shows him different techniques to master the Rasengan. This could add a whole new dynamic to the story and his character development.
Well, it depends on the specific fanfic. Some fanfics might really emphasize Naruto's growth to be on par with Minato in terms of smarts. They could show Naruto learning from Minato's memories or having some sort of spiritual connection that transfers knowledge. However, other fanfics might focus more on other aspects of Naruto's character and not necessarily make him as smart as Minato.
Well, he might start with intense training sessions. Maybe he observes Kakashi closely for the Chidori, and for the Rasengan, he practices over and over again, gradually getting the hang of the chakra control required.
Another plot device could be a mind - link or a shared vision. Maybe Naruto has a moment where he links minds with a past Hokage who was an expert in Rasengan. Through this connection, he gains instant knowledge on how to form and use it. It's a more supernatural way of learning compared to the traditional training.
The relationship can be developed by having Minato appear in Naruto's memories more vividly. For example, Naruto could start having flashbacks of Minato's training sessions or family moments that he didn't know about before. This would make Naruto feel closer to his father.
Minato might show his protectiveness by always being near Naruto in spirit. For example, he could leave behind some kind of jutsu or power that activates when Naruto is in great danger, like a hidden seal that releases a powerful defense.
Jiraiya is likely to be a popular character. He was Minato's teacher and has a strong influence on Naruto as well. In a time travel fanfiction, he could provide a lot of wisdom and help in the different situations they encounter. Also, Tsunade could be important. Her leadership skills and medical ninjutsu could be very useful when they are in the past, and she has her own history with both Minato and Naruto's family.
At first, Naruto and Kushina might start to distance themselves from Minato. They could avoid him and not talk to him as much. As time goes on, the hatred might grow deeper. Naruto could start to oppose Minato in public or in important village matters. Kushina might also try to influence others in the village against Minato.