What did the father and son expect to find when they got to the East Coast? Did they find it?

The father really felt that heading toward the sea was something. THere is no explanation of what that something might be. Perhaps the sea symbolized freedom or at least a way (by boat) to travel even farther away from the hell that they are in. Perhaps the air might not be as toxic or there might be more of a population not out to eat each other. The father feels they have to do something, anything to give his boy a fighting chance to survive.

Oh yes, they do make it to the coast. It seems just as bleak however the novel does end with a hint of optimism.