For Nate, you could develop his character by having him face his fears. Maybe he's afraid of a certain type of Pokemon, and through his adventures with Yancy, he overcomes that fear. Yancy could be developed by showing her more vulnerable side. For example, if she usually seems confident, but then she loses a big Pokemon battle and has to learn to pick herself up again.
Character development for Nate could involve him becoming more empathetic towards Pokemon. He might start off as just focused on winning battles, but as he spends more time with Yancy, who has a deep connection with her Pokemon, he starts to understand the importance of the bond. Yancy could be developed by having her face a moral dilemma, like whether to use a powerful but potentially harmful Pokemon move in a crucial battle. How she resolves this would show growth in her character.
Nate could be developed through his relationships with other characters in the fanfiction. If he meets an older, more experienced trainer, he could learn new techniques and also gain more wisdom, which would change his character. Yancy, on the other hand, could be developed by exploring her family background. If she has a family member who was a great Pokemon trainer, she could be striving to live up to that legacy and this could shape her actions and decisions in the story.