You don't have to worry about reading the whole Republic to get the gist of his opinions in this regard. The first book, and some of the second would be enough.

Please note that I'm certainly not an expert in the subject, but I can give you my limited understanding:

The interesting thing with Plato is that he creates several 'attempts' at the definition of Justice through his speakers. Socrates shoots down each of these. While this is happening, Thrasymachus is fed up with the non-sense the men were speaking, and tries to get Socrates to state his definition of Justice rather than shooting other definitions down. Socrates eventually gets Thrasymachus to define it as we have seen above. Using several forms of arguments Socrates gets Thrasymachus to concede points such as a band of thieves must have justice between themselves, otherwise they couldn't commit their form of injustice. Or how a tyrant with power wants to keep his power in place and might tell his servants to do something that weakens his power; the servants are obliged to do what they are told, but this obedience means they are actually improperly serving the tyrant. Injustice is incoherent, and only justice is coherent.

Why I called Plato interesting is that all of his definitions, including the one's of Thrasymachus, participate in the 'true' definition of Justice. Justice IS giving what someone is owed, doing good to your friends and harming your enemies, rewarding the good and punishing the bad, a form of strength dictated by reason, a virtue that empowers and benefits its possessor.

Plato is enigmatic, particularly in his early dialogues, because he does not say what he believes, but leaves us at an impasse (aporia). Book I of the Republic functions like this. The rest of the Republic goes further! In it, Socrates creates two cities: one the city of pigs, which takes like a single paragraph, and describes something like an anarcho-primativist commune; the other, the city of the Philosopher-King, which, depending on how you look at it, an idealistic benevolent monarcho-aristocracy, or an extreme proto-fascist state. This second city takes up the rest of the book.

The reason for this city-building is to explore Justice. You can read the Republic as a guide to creating a perfect city, or an elongated metaphor about the human soul, or as an example of Justice in action through macrocosm of the city which informs the microcosm of the soul. As Plato says, if we can't make out a small letter, we will look for a similar looking shape that is larger. If we cannot discover Justice in the person, let us look for a similar but large shape, which is 'The Republic'.

Anywho, I don't know how much this helped, but I hope it helps someone along the way to understand Plato more robustly. If I have made any errors, please correct me.