In some fanfictions, Naruto could find out in a very emotional way. For example, his daughter might be in danger, and when he discovers she is his, his feelings of love and the need to protect her are intensified. He might also struggle with the fact that he has missed out on part of her life so far. He'd then start to make up for lost time by spending as much time with her as possible, teaching her ninjutsu and sharing his own life experiences.
He becomes more nurturing. Naruto, who has always had a big heart, now channels that into taking care of his daughter. He listens to her more intently, tries to understand her feelings, and is always there to offer support and advice, just like a good father should. He might also be more patient than he used to be. For example, if his daughter is having trouble learning a new ninjutsu technique, instead of getting frustrated like he might have in the past, he'll patiently guide her through it step by step.
He might face distrust from Morgana at first. She could think he has been using his magic against her without her knowledge.
In many fanfictions where Naruto discovers he has a sister, it often changes the entire dynamic of his story. It could give him a new sense of family and belonging that he lacked growing up. Maybe this sister was hidden away for some reason, like to protect her from Naruto's jinchuriki status bringing danger to her. This new relationship can also influence how Naruto approaches his ninja goals and his relationships with his friends.
He could be shocked at first. Just the sudden realization that the people he thought were his biological parents all his life are not would be a huge blow.
In some fanfictions, Naruto is completely devastated. He might go into a state of shock at first, unable to believe the news. Then he could be filled with intense grief and anger, blaming himself for not being there to protect Jiraiya.
In some fanfictions, Edward might be initially shocked. He could be worried about Bella's health, considering the unique situation of a human - vampire pregnancy. He might also feel a sense of protectiveness towards Bella and the unborn child, and start making plans to ensure their safety.
He might be completely devastated. In many fanfictions, Edward enters a state of deep despair. He could isolate himself from others, feeling like his world has ended without Bella.
The sister could also bring some new conflicts. For instance, if she has a different view on how the ninja world should be run compared to Naruto. This could lead to arguments and growth for both of them. Additionally, she might have her own set of friends or allies that Naruto has to get to know and work with, which would expand the cast of characters in the fanfiction and create more complex storylines.
Naruto could also be a bit angry at first. Angry that she had been away from him all this time. But then that anger would quickly turn into overwhelming love and relief. He would have so many questions to ask her about why she wasn't there as he was growing up, but at the same time, he would be so glad to have her there now.
In many fanfictions, Naruto is initially in a state of shock. He can't believe that Jiraiya, his mentor, is gone. He might go into a period of deep sadness and isolation, withdrawing from others as he tries to process the loss.