It's quite likely Shakespeare attended the Stratford Grammar school. Though John Shakespeare was a tradesman, he was well connected in the community, and the grammar school (which is still standing) was a very short distance from Shakespeare's childhood home.

The curriculum at the time in English grammar schools was primarily based in Latin, so from an early age he would have been exposed to the works of Ovid, Seneca, Virgil, et. al., which influence one can certainly see in his later works...

A fairly extensive analysis of what he might have read can be found here, and a broader overview of his early days (which I'm drawing largely from for this answer) is here.

Although no records of his attendance survive to the present, it is very likely that William Shakespeare was educated at the Stratford grammar school starting at age 6 or 7. The structure still stands and is today known as the King Edward VI School in Stratford-Upon-Avon. The school is now in the care of the Shakespeare Birthplace Trust.

In act 4, scene 1 of The Merry Wives of Windsor, we see a Welsh headmaster testing the knowledge of his pupil who is appropriately named William. This scene is widely believed to have been based on Shakespeare's own recollections of his experiences in school. As in all Elizabethan grammar schools, Latin was the primary language of learning. Although the young Shakespeare likely had some of his lessons taught in English, he would have studied Latin composition in addition to reading Latin authors such as Horace, Cicero, Ovid, Seneca, and Virgil. He was clearly paying close attention in class, as he had an impressive knowledge of these authors as evidenced by his plays and other writings. For much more detailed information about the books Shakespeare read, please see this article by Thomas Spencer Baynes.

Based on a story told more than a century after the fact, there are some scholars who believe that Shakespeare was removed from school at the age of 13 due to his father's financial difficulties. This information comes from the notoriously inaccurate 1709 biography of Shakespeare written by Nicholas Rowe and should be regarded with extreme skepticism. Regardless of whether or not Shakespeare completed his formal education, we have every reason to believe that he had acquired a firm understanding of English and Latin by the time he was a young man.

Details of Shakespeare's life between leaving school and his appearance in London as an actor in the 1580's are impossible to trace. There are suggestions floating around that he briefly worked as a school teacher or lawyer and that he visited theaters around England and began to hone his craft during this period, but any accounts of his "lost years" should be regarded with skepticism.

William's father, John Shakespeare was a merchant and tradesman of modest (but not insignificant) means. By modern western standards we would consider him to have a comfortable, upper-middle class lifestyle. Although education would not have been compulsory, it was seen as highly desirable. Shakespeare's basic education would not have been uncommon for a person of his economic class.