As Book II focused on Socrates' beneficial teaching of self-discipline, Book III focuses on ways in which Socrates' teaching has contributed to making people better -- in particular, better in their contributions to the city.
3.1-7 Making Better Leaders. Socrates begins by discussing military matters: the education of a general (3.1), the standard for a general (3.2), the responsibility of a cavalry commander (3.3), the responsibility of a general (3.4), and, with Pericles the Younger, the goals a general should take on to make the army well-ordered (3.5).
He then is shown discussing political leadership, first arguing with Glaucon that one must only enter into politics properly prepared (3.6), then arguing with Charmides that one must not shirk the responsibility to enter into it (3.7).
3.8-14 Making Lives Well-Ordered. The second half of the book is somewhat more miscellaneous, but still, I think, has some structure. We begin with Socrates showing Aristippus the importance of a well-ordered life (3.8) and continue with various comments about Socrates' general teaching on the well-ordered life (3.9).
After this, we find Socrates talking with people in various specific occupations, so we see him contributing to the actual good order of their lives. The occupations are: painting, sculpting, and armoury (3.10) and courtesanship (3.11), or, in other words, several of the major male productive trades and the one and only productive trade available for women in ancient Greece.
Then we get some various comments and discussions by Socrates on the body: the importance of physical training (3.12) and miscellaneous comments on things like travel (which would have involved an immense amount of walking, of course), food, and drink (3.13-14).
There are several interesting things in this part of the work. Socrates' advice to the cavalry commander is closely related to Xenophon's discussion of cavalry command in his Hipparchikos. We get Xenophon's one and only mention of Plato in the discussion with Plato's brother, Glaucon. Socrates' advice is much more practical than we generally get in Plato, although it is important to note that it is not entirely without parallel -- Laches, for instance, clearly has some links with the discussions of generalship here, and there are several places in Plato's dialogues in which the importance of physical education for a life well ordered is emphasized (e.g., Timaeus, Republic). But perhaps most interesting are the discussions with Pericles the Younger (son of Pericles and Aspasia) and with the hetaira Theodote. The latter is the only time we actually see Socrates being Socratic with a woman in any extant Socratic writing. (In Plato we have several references to Socrates discussing matters with Aspasia, and Socrates' recounting of the lessons with Diotima in the Symposium, and his wife Xanthippe is technically present and speaks briefly at the beginning of Socrates' last day, but we never actually see real interaction. It is possible that Socrates interacts with Aspasia in Aeschines' dialogue Aspasia, of which we have fragments, but it's not clear from any of the surviving fragments that this is the case.)
(1) Unlike his older half-brothers, Pericles the Younger was technically born a foreigner rather than an Athenian, because at the time Athenian law confined Athenian citizenship to those with two Athenian parents, and Pericles' mother, Aspasia, was herself a foreigner. However, the Athenian assembly seems to have passed a law naturalizing him. He was elected general but had a relatively short career: he was at the battle of Arginusae, and was one of the generals executed by the Athenians due to the failure to save Athenian sailors in the aftermath of the battle.
It thus must be deliberately ironic that Socrates opens the discussion with a hope that if Pericles is elected general, the city will have military victory. They discuss the puzzle that Athens seems to have no disadvantages with respect to Boeotia (e.g., Thebes), but that Athens has not been doing very well against them (the battle of Delium, at which Socrates himself fought, is one of the examples mentioned). Socrates suggests that this might be because they are waiting for the right leader. If you were trying to get someone to take property, you could do it by convincing them that it was their inheritance, so you'd need to do the same in order to get people to reach out and take virtue. The example of taking another person's property is, I suspect, an allusion to common Athenian justifications of the Athenian empire as being the right of Athenians due to the success of the Athenians in the Persian War; Socrates contrasts this with taking the virtuous behavior of Athenians in the old stories, including stories about the Persian War, as a legacy.
Pericles wonders how the Athenians could have degenerated so much. Socrates suggests that it is negligence, as when a star athlete grows complacent through lack of effort. The Athenians could reclaim their excellence, either by rediscovering their ancestral way of life or by looking to the Spartans and working to outdo them in virtue. Pericles is skeptical of the ability of Athens to do this, given how given the Athenians are to quarrels. Socrates is more optimistic, and brings the discussion again to the importance of having the right leaders, and they end by discussing some particular things generals should do.
(2) Someone happens to mention that Theodote is beautiful beyond description, at which Socrates responds that they should go see her, since no one can get an idea from description of what is beyond description. They find Theodote posing as a model for a painter, and when the painter is done, Socrates raises a question: Should they be more grateful to Theodote for letting them see her beauty, or should she be more grateful to them for seeing it? Theodote gets the admiration, and her reputation will be spread further; but those who see her are left with only longing. Theodote notes that the consequence seems to be that she should be grateful to them.
Socrates, seeing how fine Theodote's clothes are, asks the source of her money. Theodote replies that she has generous 'friends' (philoi); and Socrates asks if she gets her friends by luck (tyche) or contrivance (mechana). This turns the discussion to hunting friends, and Socrates suggests that she should find a human hound to hunt them down for her, and then catch them in her nets. She asks what nets she has, and Socrates says she has her body, and more importantly than that, inside her body she has her mind, and the capacity to lvoe by word and deed. Socrates gives other advice for catching 'friends', and Theodote asks for his help in doing so.
Socrates jokes that he's a very busy man, and that he has a lot of 'girlfriends' (philai) who are constantly demanding his help in making love-philtres and charms. Theodote, surprised, asks if Socrates knows those, as well, and Socrates asks however else he always has his companions with him. Theodote asks him to lend her his magic wheel so she can use it first to charm him, and Socrates insists that he wants her to come to him instead. Then it ends, somewhat cryptically:
'Very well, I will,' she declared. 'Only mind you let me in.'
'Yes, I'll let you in,' said Socrates, 'unless I have someone with me that I like better.'
It is worth pointing out that a hetaira was not a prostitute; the lines between the two could blur in less reputable or more desperate cases, but prostitution was not what the hetaira did, and, like a modern geisha, to the extent that the profession itself was sexual, it was usually focused more on the sexual allure of a woman capable of crossing into situations otherwise reserved for men, rather than on sexual acts. This sharply distinguished them from actual prostitutes. Thus, despite the sexual tinge that inevitably surrounds the work of the hetaira, the discussion here isn't very far from what could be had with any businessman or consultant. (The masculine version of hetaira, which is hetairos, meant a companion or comrade and was often used of business partners. Indeed, the hetaira is in a sense a specialized business partner or consultant -- her primary task was to facilitate the networking that made business and politics possible in democratic Athens.)