Although no one seems obviously oppressed or unhappy in the opening of the Republic, as the dialogue goes on, a clear sense of discontentment with the present emerges. Some characters seem simply intellectually frustrated—they know that they're capable of more—but others, most especially Socrates, seem deeply discouraged with the state of things in general: politics, morality, religion, education, etc. Even though he doesn't describe it in this Platonic dialogue (it's in the Phaedrus), Socrates believed he was called to devote his life to spreading the truth of philosophy when confronted with the inscription "know thyself" at the Delphic oracle. How could he waste his time with anything else?
Now obviously, Plato's Republic isn't literally a quest—it's metaphorically one—so things change and develop intellectually or mentally, rather than physically. In this sense, Plato's entire definition of philosophy is a journey, one that lasts your entire life. But, to bring things back to the immediate, the micro-journey that occurs in the book would be their elaborate and difficult debate on the nature of justice before Socrates invents the republic. And you have to admit, between Thrasymachus's threats and Socrates's doubts, it's definitely a road full of dangers and obstacles.
Arrival and Frustration
After so much debating, they finally come to an agreement on the easiest and clearest way to solve their philosophical dilemma: found an imaginary city (obviously). However, even though this will ultimately prove a crucial turning point, founding the city opens up problems and consequences they never imagined. A simple resolution to their quest for the truth looks less and less possible.
The Final Ordeals
Even after they've described their city with a huge amount of detail, some big things still remain unclear. And the biggest—what is the good and what is philosophy—can best be communicated by a story called the allegory of the cave. You can get the full scoop in "Symbols, Imagery, and Allegory," but for now the important bit is that this little story is about... a quest! Yep, when Socrates gets to the big moment and has to describe what philosophy means, he tells a story about hardship, obstacles, and perseverance. Just like he and his friends will undergo all those things in order to get to the truth in this debate, so must everyone who wants to follow a philosophical life.
It's hard to believe, but this inquisitive bunch is ultimately able to come to an agreement about the nature of the good, of justice, and of an ideal city. But while there does seem to be some kind of intellectual resolution achieved by the characters, perhaps the more important resolution is that they've all witnessed, first-hand, the importance of philosophy and of critical thinking. So, unlike most quest narratives you'll read, you can actually reach this goal along with the characters.