There is conflict all throughout The Road, and seen in many different forms. In fact, the book is heavy with stress and the strain of survival in the harsh conditions created by the nature of their situation--both in the environment's nature, and in the side of human nature that surfaces in the dire times.
The first type of conflict throughout the book is man vs. nature. The father and son are constantly battling the harsh impact of nuclear winter. There is little to no sunshine, rain and cold are constant factors, and greenery has all died off and offers no relief or beauty. That climate makes sickness a constant threat, and the ability to stay warm becomes a battle every day, instead of something that just happens. Even lighting fires is dangerous, given the nature of the humans that are hunting each other. This type of conflict adds to the character development of the father; we see his struggle, and his main role of protector and provider challenged.
Other conflicts are seen in the category of man vs. man. There are two types of conflict with other humans in this novel--the first comes in trying to survive against the brutality and cannibalism that is occuring. The father becomes aggressive and vigilant in his protection of his son here. This also adds quite a bit of movement to the plot as they hide and run from these barbarians. The second man vs. man conflict is when stragglers come across their path, and the son wants to help. The father doesn't, merely for survival's sake. It is much harder to survive with more people. In these situations, we see the father's focus on survival, and the son's softer heart that is more focused on humanity's good side that is filled with compassion and mercy. This helps us to see that the son has not yet been completely tainted by the hopelessness of his situation.
Inner conflict within the father can fall under the category of man vs. self. He constantly doubts himself, his plan, and his ability to protect his son. He worries he's not doing the right thing, that he is embittering his son. He also longs for the past life, and battles that longing in the novel. He also debates helping others, deciding if survival or humanity is more important.
All of these conflicts help to move the novel forward, adds depth to the characters, and enhances the themes of survival and humanity. I hope that helps a bit; good luck!