With his friends like Sasuke, Naruto's new ability might cause some jealousy at first. Sasuke is always striving to be stronger, and seeing Naruto master such a powerful jutsu could make him feel a bit left behind. But in the long run, it could also motivate Sasuke to improve himself further and their rivalry could become even more intense.
Naruto's relationship with the village would also change. The Hiraishin no Jutsu is a very powerful and feared jutsu. The villagers might start to see Naruto as an even greater asset and protector. This could lead to more respect for him, but also more expectations, which Naruto would have to manage while still being true to himself.
His relationship with Sasuke could change significantly. Sasuke might see Naruto's newfound ability as a threat or, on the other hand, it could be a motivation for him to get stronger. Naruto might use the Hiraishin no Jutsu to try and keep an eye on Sasuke, which could either annoy Sasuke or make him secretly respect Naruto's determination.
One possible plot twist could be that Naruto discovers a hidden aspect of the Hiraishin no Jutsu. For example, it might have a connection to an ancient Uzumaki clan ability that no one knew about. This could lead to Naruto being able to use it in ways that Minato, his father, never did.
In a fanfic where Naruto learns medical jutsu, it could greatly impact the story. Firstly, it would make him more versatile in battles. He could heal his comrades on the spot, increasing their chances of survival. This would also change his relationships. For example, he might become closer to Sakura as they now share a common area of expertise. Moreover, it could lead to new plotlines. He might be sent on special medical missions or be able to develop unique jutsu combinations that incorporate his medical skills.
The Hidden Mist Jutsu could give Naruto a new strategic advantage. He could use it to cover his movements during battles. Since Naruto is often in the spotlight during fights, this jutsu would let him operate in the shadows for a change. He might be able to develop new combinations with his existing jutsus. For instance, he could use Rasengan while hidden in the mist. This would make it harder for opponents to predict his attacks and defend against them.
He might learn a time - traveling jutsu. This would be really cool as it could allow him to go back and change certain events in the Naruto world, or gain knowledge from the past to use in the present battles.
Naruto would become more confident. Learning these powerful techniques would give him a sense of accomplishment and power. He'd also be more strategic. Hiraishin is a space - time ninjutsu, and Rasengan is a powerful offensive move. Combining them would require thinking ahead. So, he'd grow in his ability to plan battles.
In some cases, Naruto could be in a really tough battle where he realizes the need for a super - fast movement technique like Hiraishin. So, he starts to search for ways to learn it. He might ask Kakashi sensei about it, and Kakashi could give him some clues which lead him to begin his journey of learning Hiraishin.
Well, in many such fanfictions, Naruto might start learning his father's jutsu when he discovers some hidden scrolls or when a mentor who knew his father decides to teach him. Maybe it could be someone like Jiraiya who reveals some secrets about his father's techniques.
If he learned a healing jutsu, he would become more nurturing. He could use this jutsu not only to heal his friends in battle but also to help the villagers. His reputation would change from just a powerful fighter to also a healer, and he would be more respected in a different way.
He could practice in secret locations where no one can see him. For example, in an abandoned cave or a hidden part of the forest. That way, no one will know about the new jutsu he's developing.