Shakespeare's work stays alive after all these centuries because it's full of amazing stories, compelling characters, and drama of everyday life told in beautiful language. However, for many modern readers it is hard to get past the archaic language and see the real story. I think one of the best ways to understand the plays is to see them performed in live theatre, and to watch films, not only contemporary versions with modernized language but also classic BBC versions that perform the five act plays in the full Shakespearean dialogue. A few of my favourites are Imogen Stubbs in Twelfth Night, Emma Thompson in Much Ado About Nothing, and Helen Mirren in As You Like It. I am looking forward to seeing Helen Mirren's new interpretation of The Tempest.
Once you are familiar with the story and characters, and can grasp some of the nuances in the conflicts and political setting of the stories, it is easier to start reading the original texts. Then Polygot's advice is good--read them a lot, over and over, and read them out loud. As you become familiar, the langauge starts to make sense.
Shakespeare wrote for performance, not for a reading public. Seeing his work performed is the best way to understand it. A reading appreciation can come later. His plays are deep and resonate on a million levels of meaning that build on a turn of a phrase. They are worth reading over and over. I find as I move through life I bring my own experience to my understanding of the plays and find resonance with different characters at different stages of life.
Shakespeare is definitely a writer not to read once and toss aside with, "Oh, I already read that." I find there is always something new to discover. Stay with it, re-read and take your time.