In any case, through most of its history Alcibiades has been regarded as the gateway to the Platonic dialogues, the one to start off with, so there are any number of reasons to read it even if it is not actually Plato's. It is also relatively funny.
You can read Alcibiades online in Lamb's translation at the Perseus Project.
(in order of appearance)
Socrates opens the dialogue addressing Alcibiades and explaining why, after having avoided Alcibiades for so long, he is now approaching him. Alcibiades has for some time been pestered by men wanting to be his erastes, but Socrates was prevented from doing so by his daemon, the divine voice that prevents him from doing things that are wrong. However, Alcibiades has repudiated most of the attempts of other men, and the divine sign now no longer prevents Socrates from approaching Alcibiades now that the latter is preparing to enter public life. Socrates offers to explain to him the real reason why he rejected the others, and why he, Socrates, can offer what they did not. Alcibiades is quite clearly not thrilled, but lets Socrates tell him, and Socrates replies that Alcibiades wants his reputation and influence "to saturate all mankind, so to speak" (105c).
Alcibiades refuses to commit on that point, but asks what makes Socrates so indispensable to that project, even if so. Socrates then begins to question him as to what he is capable of offering to the Athenian assembly. Socrates establishes that he does not know, and, as he often does, brings it around to the question of justice, which is important to public life, asking when Alcibiades learned what justice and injustice were, and how he learned it. The boy is unable to answer the question, of course, and so shows himself ignorant of the most important things:
SOCRATES: Good God, Alcibiades, what a sorry state you're in! I hesitate to call it by name, but still, since we're alone, it must be said. You are wedded to stupidity, my good fellow, stupidity in the highest degree--our discussion and your own words convict you of it. this is why you're rushing into politics before you've got an education. You're not alone in this sad state--you've got most of our city's politicians for company. There are only a few exceptions, among them, perhaps, your guardian, Pericles. (118b-c)
But Socrates in fact goes on to argue that Pericles is not an exception. Alcibiades attempts to use the fact that politicians generally go into politics without a real understanding of justice and injustice as an excuse for doing it himself, but Socrates vehemently rejects the idea, arguing that this is unworthy of Alcibiades' ambition: he is settling for competing against "Midias the cockfighter" rather than the real players on the world stage, like the king of Sparta or the king of Persia. There is then some extensive discussion of the educations of Sparta and Persia, to show that each has significant advantages over Alcibiades on important points. In order to compete with them, Alcibiades must follow the inscription at the oracle of Delphi, Know Thyself, and cultivate himself.
Alcibiades is interested by now, so the discussion turns to self-cultivation. They conclude that this is cultivation of one's soul, and in particular one must have the self-control that comes from knowing the most divine part of oneself, the part where knowing and understanding take place. This is essential to Alcibiades' political ambitions, because the real prosperity of the city lies in justice and self-control, which Alcibiades can only give to Athens if he also has justice and self-control.
The boy by this point is impressed, and insists that from now on he will approach Socrates, and be his constant attendant. He will start cultivating justice in himself right now.
And Socrates replies that he hopes that Alcibiades will persevere, but he fears that the city, because it is so powerful, might get the best of both of them.
* The complexity of the erastes-eromenos relationship is too great to get into in any detail here. It is not found in the earliest strata of Greek culture, and seems to have arisen out of the confluence of a number of factors, including a strong emphasis on athletics and a highly misogynistic culture; Aristotle famously suggested that it might have been encouraged as a means of population control. For the Greeks, however, it was a mentorship relationship, allowing for sex, apparently always intercrural, that had very strict and complicated rules attached -- even the slightest violation of the rules or customs could result in the relationship being regarded as highly vulgar or degenerate. The relationship between Socrates and Alcibiades was famously nonsexual and purely educational; this is, in fact, the root of our phrase, "Platonic love".
* The four cardinal virtues make a brief showing in the description of the education of Persian princes (121e-122a).
Although the argument is fairly straightforward, this dialogue covers quite a few issues. There is an especially interesting discussion of what the Delphic inscription really means. This includes a fascinating argument that part of knowing oneself is dialogue with another:
SOCRATES: You think about it, too. If the inscription took our eyes to be men and advised them, "See thyself,' how would we understand such advice?Shouldn't the eye be looking at something in which it could see itself?
SOCRATES: Then let's think of something that allows us to see both it and ourselves when we look at it.
ALCIBIADES: Obviously, Socrates, you mean mirrors and that sort of thing.
SOCRATES: I'm sure you've noticed that when a man looks into an eye his face appears in it, like in a mirror. We call this the 'pupil', for it's a sort of miniature of the man who's looking.
SOCRATES: Then if the soul, Alcibiades, is to know itself, it must look at a soul, and especially at that region in which what makes a soul good, wisdom, occurs, and at anything else which is similar to it.
ALCIBIADES: I agree with you, Socrates.
SOCRATES: Can we say that there is anything about the soul which is more divine than that where knowing and understanding take place?
ALCIBIADES: No, we can't.
SOCRATES: Then that region in it resembles the divine, and someone who looked at that and grasped everything divine--vision and understanding--would have the best grasp of himself as well. (132d-133c)
If I had to pick one thing that sums up the dialogue, I think it would be this argument that knowing oneself requires interacting with someone else so as to recognize what is divine in them. Not only is it argued for, it is, in a sense, what Socrates has promised from the beginning, and what he is beginning to model throughout the dialogue. It also explains, I think, why Alcibiades is so genuinely intrigued by Socrates' offer.
Quotations are from D. S. Hutchinson's translation in Plato, Complete Works, Cooper and Hutchinson, eds., pp. 557-595.