He never went to university. He was on track to, but his father, John, had financial problems, so was forced to pull his son out of school to work in the shop. Had Shakespeare finished school, he would have gone on to either Oxford or Cambridge university and read one of three sciences, Law, Medicine or Theology. He had had a thorough education, which revolved around Latin, to which he was introduced at the age of nine. The tutors at his school were all Oxford men, (Simon Hunt, Shakespeare's tutor, went on to become a Jesuit). Shakespeare was given a book by his mother, Mary Arden, to use as a Latin 'crib'. Ovid's 'Metamorphoses', a book he kept at his side all through his life, indeed so influential was this book, that he included it in 'Titus Andronicus'. His reading was phenomenal, taking in, amongst others, 'Lives of the Noble Grecians and Romans' by Sir Thomas North, a book which Shakespeare regularly dipped into for inspiration in his 'Roman' plays. Works in French and Italian, ( he could read both). Interpretations of old Latin texts, giving him confidence to write fresher versions, and an 'understanding' of what was going on in England at the time, promoting such works as 'Julius Caesar' and 'Coriolanus'.
There is very little known of his childhood, and no evidence that he attended a formal school. That is part of the evidence that someone else actually wrote the works attributed to him. Obviously whoever wrote it had an excellent knowledge of classical literature.