Our hero doesn't exactly start his ordeal voluntarily. He's "selected," as they say, to bear the burden that the rest of the community would rather not deal with. But we have to remember that Jonas has some innate qualities which, along with a pair of baby blues, convinced the Elders he was the right man for the job. In a way, it's almost as if he was fated.
This part of the Booker plot is made up of difficult ordeals faced by the hero. Every tough memory which Jonas encounters is another ordeal. He has to suffer through it physically, sure, but also emotionally. As he grows farther away from his friends and family, Jonas has to deal with isolation as well, and a (justified) sense of mistrust.
Jonas arrives "in sight of his goal," as Booker says, when he and The Giver think it may be possible to change the way things work in the community. When Jonas finds out that Gabriel is to be killed, this is the last obstacle in sight.
Jonas struggles to keep Gabriel and himself alive even while in sight (or so he thinks) of his end goal: Elsewhere. The "final battle" is of course that last uphill climb, though no one can tell you for sure whether Jonas makes it alive to his goal. Call it an ambiguous "Quest," but that's the way it goes.