One conflict is man vs. self. This conflict is Winnie's conflict. Should she accept the offer of immortality or not? Only she can determine that for herself, and she struggles with it quite a bit. The reason that she wants to take the immortality is because she wants the freedom that comes with it. To go off with the Tucks, she would not be under the rules and regulations of society. She reveals this desire early on, when talking with the toad:
"It'd be better if I could be like you, out in the open and making up my own mind."
However, what she comes to understand is that too much freedom is dangerous. Without limits, there is danger. Also, without a limit to life, joy loses is power. Winnie comes to understand that love and joy are so intense because of the mortality of humans. We have a short time on this earth, and must cherish our happiness. Take away time, and you take away happiness. Jesse's parents make this clear to her, and she bases her choice on their advice.
Another conflict is man vs. man. The greed of the stranger in the yellow suit threatens the Tucks and Winnie. He wants so much to have immortality that he is willing to reveal the Tucks to society. They would be victims of any number of strangers seeking to have and to understand what has happened to the Tucks. This is why Mae is forced to murder the man, ending the conflict with him.