For the average American high school student, your experience with Shakespeare may have consisted of the following: being handed a beaten up copy of Hamlet or Romeo and Juliet, leaving it in the bottom of your locker for three weeks, and reading the SparkNotes page the night before the exam on it to cram all of the details about the characters, language, and subplots in one sitting. If this has been your one and only (painful) experience with a Shakespearean play, I am sorry.
There are several common grievances against the playwright, mainly being that it is too boring, too old, or too complicated to understand.
For starters, yes, it is complicated. Reading elevated language is an acquired skill that must be taught in the classroom and a large part of it is plain experience with the syntax and vocabulary used. Once you read Shakespeare, each work that you read after that becomes easier and easier to understand, and significantly more enjoyable.
In terms of plot complexity and dynamic characters, that is often what makes a work of literature so enjoyable. J.K. Rowling created an entire world and dozens of main complex characters along with mentioned characters that total up to the hundreds range, yet the Harry Potter series is still wildly successful among young children. After comprehending Shakespeare, it could put a cheap love story to shame.
In terms of relevancy, Shakespeare is frequently present in pop culture, even if you don’t notice it. The archetypes that he created with his literature are so strongly rooted in our culture that people often don’t even notice anymore. Don’t believe me? The idea of star-crossed lovers is one of the most common tropes, and can be seen in works such as The Hunger Games, Titanic, and even The Little Mermaid.
Besides the famous love story, his works cover many of the aspects of human emotion that we appeal to on family relationships, historical events, comedy, and postcolonial conversation. Literature often articulates the thoughts that we cannot put into words, and regardless of the passing of centuries, Shakespeare is timeless.
On top of this, Shakespeare helped shape the English language as a whole. He invented countless words that we use on a daily basis without giving it a second thought. Even though there are far too many to list, you might take a second to remember Shakespeare when you use the words 'excitement', 'flawed', 'monumental', 'impartial', and even 'swagger'.
All in all, the works of Shakespeare may be difficult to grasp at first, but are still a monumental part of modern literature and the English language we use today.
(A list of more words that Shakespeare coined can be found here.)