William Shakespeare never actually provides stage directions that describe the appearance of Romeo. It is actually uncommon for playwrights to give detailed physical descriptions of characters, as this would make casting the parts more difficult. There are a few subtle clues as to what his appearance might be in several lines of dialogue which we will examine here. However, most descriptions concern Romeo's personality and characterization, not his physical appearance. Therefore, what Romeo looks like is mostly left up to the reader to imagine or the casting director to decide.
We can assume that Romeo is a good-looking young man. Juliet certainly thinks so, and the Nurse agrees. In fact, the most thorough physical description of Romeo comes from the Nurse in act 2, scene 5:
Well, you have made a simple choice. You know not how to choose a man. Romeo! No, not he, though his face be better than any man’s, yet his leg excels all men’s, and for a hand and a foot and a body, though they be not to be talked on, yet they are past compare (2.5.38–43).
Although there is nothing specific mentioned here about Romeo's appearance, it is enough to tell us that, at least in the opinion of the Nurse, Romeo is one of the best-looking men in Verona.
Juliet certainly agrees with the Nurse's assessment. However, she never goes into as much detail. The best description that Juliet gives of Romeo is to comment on his perfection in act 2, scene 2.
So Romeo would, were he not Romeo called,
Retain that dear perfection which he owes
Without that title (2.2.45–47).
Juliet may mean this perfection to apply to Romeo's appearance. She is certainly smitten by him and his looks, which are perfect as far as she is concerned.
When Romeo crashes the Capulet ball in act 1, scene 5, Lord Capulet also shares his opinion of the young man. His description pertains more to how Romeo comports himself as the gentleman of noble birth that he is rather than a description of his appearance.
He bears him like a portly gentleman (1.5.65)
Although Romeo has come to his enemy's party uninvited, Lord Capulet has decided to let it slide because Romeo behaves and appears appropriately. By this short description, the reader/audience can assume that Romeo is well dressed and appears as one would expect someone of nobility should.