I guess the main conflict in the book is boy vs nature, Sam struggling to find a way to survive in the woods without help. Sam has many adventures learning how to live -- cooking fish, burning out the tree for a house, climbing after crow eggs, or braving the attack of a falcon to steal a falcon chick to train. Other adventures have to do with avoiding or dealing with hunters and other people who stumble upon him or come looking for him. Later, the conflict becomes internal -- after proving to himself that he can survive, the question becomes whether he should keep doing it or go back to civilization. This conflict is resolved for him, however, when his family decides to join him in the woods.
There is another way to think about the conflict though, which is to consider the question of self-reliance and whether it is possible to be truly self-reliant. In this sense, the conflict is between Sam, and his desire to "prove" himself, and his father, who gives Sam permission to go to the farm, but, in the end, never fully lets go. To the extent that Sam is always living in opposition to, or in relationship to, his family and civilization, he is always dependent (psychologically) on others. In other words, his struggle is to change into something other than a boy from New York; that he is never fully able to do so is suggested by the "happy ending," when he is reunited with his family.