From numerous references in the play, we know that Romeo can read and write. During the balcony scene, he notes, "Had I it written, I would tear the word," referring to his family name. Later in the play, the Friar indicates that he and Romeo (now banished to Mantua) will communicate by letter, and it is a letter, sent by means of another friar, that goes amiss, causing Romeo to remain unaware that Juliet's death is being staged. Young men of Romeo's class would also usually have some schooling in philosophy, mathematics, Latin (possibly Greek too), music, dancing, and manners.
As for Juliet, it was less traditional for upper-class girls of this time and culture to be as formally educated. We have no evidence in the play that she can read and write; she sends the Nurse to personally deal with Romeo, and there is no note used. But, on the other hand, Shakespeare gives her language, especially in her famous "Gallop apace, you fiery-footed steeds" soliloquy, that indicates that she is familiar with mythology, so it is reasonable to assume that she has been educated; and indeed, it would be customary for her to at least be able to read and write. Some women, including an example right in front of Shakespeare, Queen Elizabeth I, were very highly educated and could read, write, and speak several languages, as well as dispute with scholars on obscure and complex religious matters. Juliet (as well as Shakespeare's other upper-class female characters) is clearly a learned individual.