The Italian Renaissance saw a fourishing of lierature, the sciences, and the arts. It should not be forgotten that the Middle Ages had it's own Renaissances, which not only kept alive the philosophical and literary traditions of Ancient Greece and Rome, but also contributed its own works to European culture and learning.
The seeds of the Italian Renaissance of the 14th-16th centuries are traced to the 14th century poet Petrarch's discovery of the works of the Roman Cicero. It is also notable that Petrarch also included the Roman epic poet Virgil and Roman playwright Seneca as major influences.
As the Romans kept alive the Greeks, and the Universities of the Middle Ages kept both the Greeks and Romans alive, so to then did poets, writers and scholars of the Italian Renaissance keep the traditions of Greek and Roman philosophy, literature, sciences, and arts alive in their study and in the production of new works. The English Renaissance drew hevily from classical Greek and Roman sources as well as European literature.
It's worth remembering that Europe traded with the Far East by way of the Silk Road that ran from China through central asia to the mediterreanean world. The ancient Greek historian Herodotus gives details of ancient India in his Histories. The book of Marco Polo's life and travels, Book of the Marvels of the World, known more popularly as The Travelled of Marco Polo, was published in the year 1300. It provided a wealth of information to Europeans about Asia. Many other Europeans travelled to Asia during that time.
A continuum exists that begins with the Ancient Greeks and continues throughtout the Roman Republic and Empire, kept alive during the Middle Ages in Europe and England by Universities and the Catholic Church, and continues its way to the Italy of the Italian Renaissance and continues its way to 16th cneutry England. Europe shares it's culture and literature with England, and it would be the works of ancient Greece and Rome which the English would draw upon to create their own Renaissance during the 1500s.
The Middle Ages had a flourishing culture. They saw the birth of Europe's oldest Universities. Learning, study, literature, and the arts were kept alive in these centers of learning. The Muslim world also contributed alot to the arts and sciences of the world. For the sake of Shakespeare, it's important to remember the richness of the world's cultures from ancient times to his own.
The European Middle Ages had its share of philosophers and theologists. They also saw the publication of the Canterbury Tales by Chaucer, Dante wrote his Divine Comedy, and many tellings of the Aurthurian Legend. The most famous, Le Morte D'Arthur by Thomas Malory, was published in 1485 by English bookprinter William Caxton.
The English authors and playwrights of the English Renaissance of the 1500s created works in their own style based on the works of others from Europe, both their contemporaries and the ancients. They studied the Greeks and Romans either as students of English Grammar Schools or at Universities. They wrote plays on ancient characters, fictional and real-life historical figures, such as Queen Dido, Julius Ceasar, and his general Anthony and Cleopatra, the Queen of Egypt. They also wrote tragedies in the Greco-Roman form. They also wrote History plays about their own Kings, just as Greeaks created plays based on the events of the Trojan War, the history of other famous Greeks.
Shakespeare adapted style and stories from the ancient Romans Seneca and Plautus, as well as contemporary Italian writers who had been translated into English. Shakespeare was also influenced by his fellow Elizabethans. Just as Greek playwrights adapted stories from Greek mythology and from Greek history, so too did Shakespeare adapt stories from history and literature for the English stage.
All of the playwrights of Elizabethan England started by performing the plays of the Greeks and Romans. In time, they wrote and performed their own plays, their own histories, tragedies, and comedies. Genres were also "invented" in the 1590s such as the Domestic Tragedy.
The Italians of the Italian Renaissance depended and drew upon the ancient Romans. The ancient Romans playrights created in their plays their own versions of stories portrayed in Greek dramas, and Roman Comedies were in the style of Greek New Comedy. In the same way, Elizabethan playrights adapted and borrowed traditions, forms, and styles from European sources from the ancients to their contemporaries.