Internally, Montag struggles with the realization that he is not happy. He is dissatisfied with the way that his life is, with the lack of a relationship with his wife, and with his society. He struggles between conformity, and seeking out change. He battles this within himself for most of the novel, and that battle is represented symbolically by his criminal hand that seems to have a mind of its own, stealing books and hiding them in his house. He wants to change his life, but doesn't know how. He doesn't know where to start, or even what really needs changing. So he struggles to find answers in books, and goes to Faber for guidance.
Externally, Montag experiences conflict with his wife; she denies being unhappy or having attempted suicide, she defends her airhead friends and t.v. shows, and is irritated with his attempts to read with her. He wants to change, and wants her to come along with him, but she doesn't want to go. He experiences conflict with Beatty, as he delves further into understanding books and why society is the way that it is, Beatty is there the entire time, watching him, baiting him, taunting him, and trying to push him back in the other direction. He experiences external conflict with his society; he is at odds with what his society says is "proper" behavior. This conflict goes full-blown after he kills Beatty and has to escape from his society in totality.
I hope that those thoughts help a bit, and good luck with everything!