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.

We know that Romeo could read and write.  In the second scene of the play, when the illiterate Capulet servant approaches with the guest list to the party, he asks if Romeo can read:

PETER: Perhaps you have learned it without book. But I pray, can you read anything you see?

Romeo is a bit demeaning when he says:

ROMEO: Ay, if I know the letters and the language.

Though he is intending to be flippant (a result of his attitude over Rosaline), here Romeo suggests that he has been formally educated in reading and writing, and perhaps knows more than one language.

It is likely, then, that both Romeo and Juliet were formally schooled in the basic subjects of reading, writing, and mathematics.  In addition, they are likely also privately tutored in the arts, things like painting or singing, as well as fencing, dancing and etiquette.

The formal and personal education the two likely received would also contribute to the idea that both Romeo and Juliet were sheltered and spoiled.  This helps to explain their lack of sound judgement in matters of love and relationships, as well as their hasty decisions leading them to getting what they think they want immediately.