Andrew Sullivan links to an article in the Guardian featuring the thoughts of five authors on Shakespeare the man. In that spirit, I wish to add my two cents (pence?, not here in the States) worth.
Back when I was at UCLA, all engineering students had to take three upper division humanities courses within a single department or on a theme (Italian history, art and literature would be a good example). I wanted to take three semesters of Shakespeare. I got two down, but the third wasn't offered until I was due to graduate, so I ended up taking a course in detective fiction, which was surprisingly good in its own right.
Back from the digression, I really got started in following Shakespeare back in high school with the obligatory memorize-a-soliloquy assignments. That year is was Julius Caesar. Almost everyone did the funeral oration (Friends, Romans, Countrymen, lend me your ears...), while I went with Antony's lamentation over the body of Caesar. The next year, our English class was given passes to see Mel Gibson's production of Hamlet. Not bad, most definitely a envisioning for a mass audience. Kenneth Branaugh's version a few years later was more cerebral, which was good because after four hours or so, you weren't feeling anything below the neck.
The big attention grabber for me was participating in a production of As You Like It. I saw the flier announcing auditions and thought, "What the hell, I don't have a job, let's go for it." I ended up getting the role of Silvius, which in this production called for some physical comedy. I'm still looking for a practical use for the skill of drooling on cue that I developed for the scene involving Phebe's contemplation of Rosalind's male alter ego. No luck so far.
My favorite part about the writing of Shakespeare is how he wraps the meaning of words around one another and guides the reader to precisely the symbolic interpretation he desires. Of course, I could be mistaken, as many people seem to get different reads than I do. Still, Shakespeare's writing, especially the sonnets, seem to play with meanings more cleverly yet clearly than any other author I have studied. The need to be able to comprehend both line and totality adds levels to his works that make them constantly new. That is Shakespeare's genius, he doesn't need to be alive to keep coming up with new stuff, because his existing works can become something else with just a shift of perspective.