Which word for you is more "correct" to use when beginning a line (from my sonnet) that reads "[You] fairest girl e'er..."?
Good question! The distinction between, "thou" and "you", the two forms of "you" in Shakespeare's English is more or less the same as the difference between "du" and "Sie" in modern-day German.
"You" is the more formal, more "correct", more polite version ("Sie" in German). "Thou" is the more familiar version, which you'd probably never call a social superior, but that you might call your friends. "Thou" can also be an insult, as the following injunction from Sir Toby to Sir Andrew in "Twelfth Night" suggests:
'if thou thou'st him some thrice, it shall not be amiss" (3.2.50-51)
So, in short, it depends whether your girl is a social superior. If it's a love poem to your teacher, or someone you've only seen in a painting, I'd use "you". If it's a friend, someone you already know intimately, or a current girlfriend, "thou" would be better.
The difference between "thou" and "thee" is between nominative and objective. "Thou" does verbs, and "thee" has verbs done to it.
So if your sentence was, "You are the fairest girl ever", you'd need "Thou art...". If your sentence was "I think you are really beautiful", you'd need "thee" ("I think thee right fair..." or whatever).
Incidentally, I'm not sure what sentence could possibly begin with "[You] fairest girl e'er..."... I'd love to read the sonnet!
Hope it helps!