The physical danger that the father and son face as they embark on their odyssey along the road feature the gangs of men that roam this dystopian landscape where nothing grows, looking for other humans to kill and eat. Early on in the story, the father and son see a group just like this as they are on the road:
The could hear the diesel engine out on the road, running on God knows what. When he raised up to look he could just see the top of the truck moving along the road. Men standing in the stakebed, some of them holding rifles. The truck passed on and the black diesel smoke coiled through the woods.
The biggest physical danger for the father and son therefore comes from other humans and the way that humanity has regressed back into a pack mentality where packs feed off other packs and pick of lone survivors of whatever disaster has struck the planet.
As regards psychological dangers, it is clear that the biggest psychological danger is despair. This is of course represented in the boy's mother, who, facing an uncertain and dangerous future, chooses to give in to despair and kill herself rather than try and ensure survival. This is a pressure that hangs over both central characters throughout the novel and that we are reminded of on their journeys as they discover, for example, the house where the family hung themselves rather than face the future.