In "By the Waters of Babylon" who are the two main characters? What problems do they face and how do they solve them?

There really is only one main character, and that is John.  He narrates the story, and is the main character enacting all of the action.  We hear the events from his viewpoint, and get his thoughts and takes on everything.  His father also plays a role in the story, but it is more of a minor one.  We meet his father at the very beginning as he sends John off on his spirit walk, and then at the end, as he dispenses valuable advice to John about what to do with the great and overwhelming knowledges he has gained in the city of the gods.  So, John is the main character, and his father is also a key figure in the story.

Another character that is integral to the meaning and depth of the story is the man in the house that John discovers near the end of his journey in the ruined city.  The man is dead, but John is struck by his seeming calmness in the face of the destruction of his society; he is just sitting there, seemingly accepting of what has happened.  The man has a lot of books and artwork in his home, and John senses that he was very wise and knowing.  Though not a live, active character in the story, he does play a role in impacting John's opinions of what happened.

The main issues that the main character, John, faces are those of conquering his fear of the unknown lands, his desire for further knowledge, and once he has that knowledge, what he should do with it.  He takes his journey and pushes out beyond the accepted boundaries of their land, breaking ancient rules and traditions as he visits the place of the gods in search of further knowledge.  He discovers quite a bit, and takes it home to his tribe, and plans on giving that knowledge out bit by bit so that they can use it wisely.

I also provided links to the story and some great discussions and background on it that I think that you will find helpful.  Good luck!