I think it is a rather bizarre suggestion that one should read Plato in the order believed correct in antiquity. For one, it seems to me that the ancients were incorrect about that order. And, as you say, there are several dialogues the authenticity of which are in doubt and so should probably not be read before other (usually more central) dialogues the authenticity of which are not in question.
In terms of order, you can do worse than begin the five traditional "death of Socrates" dialogues. These are the Apology (Socrates' trial speech and the nature of the philosophical life), the Crito (the nature of law and implicit political contracts with the state), the Euthyphro (the nature of piety), the Meno (the nature of virtue and knowledge), and the Phaedo (the afterlife and universal objects).
These five dialogues are from different periods in Plato's writing: the Apology is probably quite early and is a good introduction to Socrates; the Euthyphro is a nice "what is x?" dialogue which comes to no resolution, which is typical of early one-on-one exchanges, and although fictional might be the sort of thing the actual Socrates went around doing; the Crito is probably a late-early dialogue, and mixes in some more serious concerns about the nature of the state that will capture Plato's attention in his later writings; the Meno starts out as a "what is x?" dialogue, but the second half transitions to consider the nature of knowledge and how we can know anything (if at all); and the Phaedo is a full-blown middle-period dialogue where Plato deploys his account of universals called the Theory of Forms (which he arguably discards or submerges in the late period, but is generally what people think of when they think of Platonic doctrine).
After the Phaedo, I would hit the other big middle-period dialogues that are classic Plato: the Republic, the Symposium, the Phaedrus. The Republic may very well be the greatest piece of philosophy ever written; if not, it is up there in the top five-to-ten. It ostensibly deals with the nature of justice, but it covers huge areas including knowledge, beauty, truth, education, and more. It develops some of the themes from the Phaedo. The Symposium is a collection of beautiful speeches on the nature of love, again developing some of the themes of the Phaedo, but also the Republic. The Phaedrus looks at the nature of love and the nature of rhetoric, picking up themes from the Republic and the Symposium.
That's a lot of stuff. If you want one and only one thing to read that's short, go for the Apology. If you want one and only one thing to read that's all-encompassing, go for the Republic.
