God, my favorites change constantly, but Hamlet, Macbeth, Romeo and Juliet, Richard III, Antony and Cleopatra, Twelfth Night, and Much Ado About Nothing all strike me as good entry points.
But also, reading Shakespeare is great, but watching it is even better. If there are local performances, try to see some. Twelfth Night is one of the ones I think gets performed the most around here, and is a great first Shakespeare for a lot of folks. The Kenneth Branagh films of Henry V and Much Ado About Nothing are also a blast (and the contemporary Joss Whedon take on Much Ado is surprisingly fun, too).
Then start enjoying how the plays get interpreted and reinterpreted. There are modern takes, sci-fi ones (Forbidden Planet is a twist on The Tempest), etc. There's a fantastically funny tv show called Slings and Arrows about a theatre company in Canada, with the seasons devoted to productions of Hamlet, Macbeth, and Lear, that I just love (but it works best once you know the shows).
Forgot about Forbidden Planet! Leslie Neilson playing it straight…. surely you can’t be serious :)
Thanks for all the show n movie recommendations! Singapore Repertory Theatre puts on its adaptation of a Shakespeare play at a public park every two years. I was lucky to watch Julius Caesar last year.