The Blue Wizard of Oz asks, "Can you recommend, for someone mostly unfamiliar with Shakespeare, a good place to start to appreciate his prose or poetry?"
Good question. I don't know. I just found out I won't get a chance to teach my Shakespeare, science fiction or film history classes next fall, so I appreciate the change to ponder Uncle Bill.
Going to see quality productions of Shakespeare seems like a good start. I love how after I see a Shakespeare production I think in iambic pentameter. You could also watch some good Shakespeare movies. (What constitutes a "good Shakespeare movie"? Well, I love Orson Welles' Macbeth, Chimes at Midnight, and Othello, but I think they all bored me when I first saw them. I love Olivier's Hamlet, Henry V, and Richard III too. I remember seeing part of Olivier's Hamlet around second grade, and I liked the ghost and the poison in the ear. Snobbishly I tried to memorize the "To be or not to be" soliloquy.) (You haven't really appreciated Hamlet until you've read it in the original Klingon.) (Just kidding.) I also like the 30's version of A Midsummer's Night Dream with Jimmy Cagney and Mickey Rooney. I also love Kurosawa's versions of King Lear (Ran) and Macbeth (Throne of Blood).
For the poetry, I remember reading comedies in a group of friends in college. We read A Midsummer's Night Dream sitting on the grass, and we read Twelfth Night in my dorm room. Some folks at the Maybe Logic Academy had talked about starting a Shakespeare study group. I also like two anthologies of Shakespeare's poetry Ted Hughes edited. I think you can easily get one called Essential Shakespeare.
I recommend getting a complete Shakespeare and just reading out loud.