I'll be reading the Platonic dialogues (including the spuria) this summer, and have already started getting some of the spurious dialogues out of the way to leave more room for the rest. (The Definitions, Halcyon, and Sisyphus have already been done, and this week I should have Demodocus, Eryxias, and possibly Axiochus done, leaving only forty-nine more to go!) Several people expressed interest in reading along for June, so I thought I'd say something about my current thoughts for how it will go and give some relevant links.
Alcibiades (on self-knowledge), Rival Lovers (on the relation between learning and philosophy), and Charmides (on self-control or temperance) are first up. None of the three would be considered especially important today, but together they make a very good introduction to the Platonic dialogues as a whole -- Alcibiades, in fact, was used for centuries by the Neoplatonists for precisely that purpose. Of these works, Charmides is certainly genuine. Plato scholars are divided on whether Alcibiades is genuine; if it is not, it is nonetheless historically the single most important and influential spurious dialogue. Rival Lovers is usually thought to be spurious.
Since people may be reading along as they can, I want to hit some of the major dialogues, without planting them so thickly that it's difficult to keep up. Plato's masterpiece, the Republic, is a super-dialogue, twelve books long, and when people counted Plato's dialogues in antiquity, they sometimes counted it as twelve dialogues. It would crowd out other dialogues, and I'm not planning on doing it in June. The next obvious idea would be to do the Last Days dialogues -- Euthyphro, the Apology, Crito, and Phaedo. However, I'm planning on mixing these with Xenophon's Memorabilia and Apology, so I'll be saving those for later, as well.
When these are out of the way, the five dialogues that are most influential historically are the Symposium, Phaedrus, Gorgias, Meno, and Timaeus. The Symposium I'm keeping for later for the same reason as the last days -- I want to pair it with Xenophon's Symposium. So that leaves four. So my current thought is that I will go through them in this order (with very roughly a week to each):
I. Phaedrus (on rhetoric, philosophy, and love), which I will pair with Ion (on poetry and rhapsodic recitation of poetry) and Menexenus (on recitation of rhetorical speeches). All three are generally considered genuine; doubts occasionally get raised about Menexenus, which is a very atypical dialogue whose purpose is unclear, but Aristotle appears to have regarded it as Plato's, and that's about as strong an external evidence of genuineness as a Platonic dialogue can get. Phaedrus is the must-read of the three, though.
II. Gorgias (on rhetoric, justice, education, and philosophy) is the one that I personally would regard as the absolute must-read among all Platonic dialogues. I haven't decided for sure what other dialogues to go with it, but I am leaning toward Lysis (on friendship), Laches (on courage or fortitude) and On Justice. The last is spurious, but the others are all genuine dialogues.
III. Timaeus (on the order of the world) is a very different kind of dialogue, but it is genuine and historically is perhaps the most influential dialogue Plato ever wrote. Timaeus has to be paired with its unfinished sister dialogue Critias (on order in society); together the two dialogues are the source of the Myth of Atlantis. I might also add Philebus (on the good of human life). All three of these are genuine.
IV. Finally, toward the end of June if it happens, or else into the first week of July, Meno (on whether virtue can be taught), along with On Virtue, Clitophon (on pursuit of virtue), and probably Euthydemus (on sophistry and philosophy). (I've partly saved the Meno for last because I know Enbrethiliel's already read it!) Of these Meno and Euthydemus are genuine, On Virtue is spurious. Clitophon is the single biggest mystery in the Platonic canon. It could be genuine; ancient authors insist, sometimes vehemently, that it is genuine; and yet the dialogue consists almost entirely of an attack on Socrates and his methods. Is it a satire of Socrates's critics? Is it spurious, an anti-Socratic dialogue that accidentally wandered into the Platonic canon? Nobody knows, and scholars are still divided.
This is all a bit up in there, and parts still might be changed (and depending on what time I have I might also throw in several extra spurious dialogues), but this allows people flexibility to decide how much they want to commit. I would recommend Alcibiades and/or Charmides, Gorgias, Phaedrus, at least the beginning of Timaeus along with the Critias, Meno, and (if we end up doing it) Philebus. As a solid minimum tour. I've put the relative lengths (in Stephanus, or standard edition, page numbers) below, so people can get some sense of the reading burden.
Alcibiades -- 32
Charmides -- 23
Gorgias -- 80
Phaedrus -- 52
Timaeus (complete) -- 75
Critias -- 15
Philebus -- 56
Meno -- 30
But, of course, Plato is interesting whatever dialogue he is writing!
Alcibiades, (Rival) Lovers, Charmides, Lysis, and Laches are here.
Euthydemus, Gorgias, and Meno are here.
Ion, Clitophon, Menexenus, Timaeus, and Critias are here.
Philebus and Phaedrus are here.
In all these cases the table of contents is on the lefthand side of the page.