Sadly, for a man of William Shakespeare's stature in the literary world, there is very little documented information about his personal life. There is no specific proof that Shakespeare ever had an affair outside of marriage, and there is likewise no documentation that he dabbled in homosexual encounters.
Most Shakespeare scholars will point to several examples in his writings that may hint toward these questions. There is the mention in Shakespeare's will leaving his wife, Anne Hathaway, "the second best bed with the furniture." Many people interpret this to mean that Shakespeare's best bed was meant for someone else. However, it is likely that he meant it in the strictest literary sense--since only one-third of his possessions lawfully belonged to the wife, it is possible that the "best bed" was merely bequeathed to another relative or friend.
Many of Shakespeare's sonnets--at least 26--are addressed to a married woman known as the "Dark Lady." Whether Shakespeare had an affair with the "Dark Lady" is only conjecture; the sonnets themselves may be strictly fictional. A larger number of his sonnets are addressed to a "Fair Lord" or "Fair Youth," but again, no proof or identification of the recipient is given. The sonnets are open to interpretation, and most of them concern the nature of physical love written in a 17th century style that can be quite ambiguous to the modern reader.
Scholars are divided on the issue of Shakespeare's sexuality and whether or not he had affairs. There are very few primary source documents available, and those that are accessible are debatable. For instance, some scholars suggest that the relationship between Shakespeare and Anne Hathaway was not good using the evidence of his will, in which he left her the "second best bed" and some other furniture. Others purport that in Shakespeare's day, it was common for the best bed in the house to be reserved for guests, and the second best bed was often the marriage bed. So you see, it's open to interpretation whether that was a slight or a nostalgic gift. For your assignment, you will need to make a claim (argument) and support it with evidence. I can give you some sources to help, as well as some background on the issue.
Shakespeare married Anne Hathaway in 1582. She was 26 and he was 18. Six months later, she gave birth to a daughter, Susanna. Of course, it can be inferred (but not proven) that they were married because of this pregnancy, since it is obvious from the dates that Anne was pregnant before her marriage. Two years later, she gave birth to twins, a son and a daughter. A year after that, Shakespeare left his family in Stratford-upon-Avon and went to London to work.
When Shakespeare moved to London, he was able to work as a playwright without having any other employment thanks to his patron, the third Earl of Southampton, who financed everything. There are surviving documents that Shakespeare wrote to or about Henry Wriothesley, the Earl, and they are understandably grateful in tone. Shakespeare was indebted to him. Some scholars speculate that the "Fair Lord" and the "Fair Youth" Shakespeare wrote about in the sonnets (particularly sonnets eighteen and twenty) are written about the Earl, and most agree that they contain evidence of homoerotic love.
The identity of the "Dark Lady" Shakespeare speaks of in the sonnets—and whether or not she was a real woman with whom Shakespeare had an affair—remains unclear. John Manningham, a contemporary of Shakespeare's, suggested in his diary that Shakespeare had an affair with a woman during a performance of the play Richard III.
Below, I have provided links for you to gather more information and to use as references for your assignment.