One of the most significant symbols is that of the fire that the father and son "carry" with them. The father tells his son over and over that they carry the fire, and the son asks about it often,--"Do we still carry the fire?". This is not a literal fire; in fact, they have trouble starting fires in the damp, ash-ridden world that they live in. It is a symbolic fire representing humanity's decency, kindness, goodness, morals and beliefs. The world that they live in is a result of war, of evil, of cruelty and desecration of any semblance of goodness. And, all that they encounter in their world is barbaric, depressing and uncivilized. So, the fire the father speaks of is a symbolic fire--the light and warmth of good, decent, human love and happiness.
Another symbol itself could be the environment--the constant rain, cold and ash that falls and surrounds them symbolizes the darkness of a world without hope, without decency, and without kindess. The setting that this boy and his father live in is so merciless, so unrelenting, and so hopeless that it is very depressing, just as a world that has no good in it would also be hopeless, dark and cold.
The father also mentions "godspoke men" and how they are gone from the world. This is a symbol of any people that will represent goodness and morals. It isn't a literal man who speaks of God, but people who represent the virtue, integrity and upstanding behavior that typically went along with a god-fearing society. That is gone now, and so are all of the "godspoke men."
I hope that those ideas give you some things to think about; good luck!