Stratford-upon-Avon is often called a must-see because of the "charming English village" image it maintains. There are many towns like this in England, but because Shakespeare was born here, it gets more tourists and is more widely known. (Plus, you can wander around gawking and the locals are used to you.)
Aside from saying "eh, Shakespeare was born here," other W.S. attractions include a Shakespeare library where you can check out just about any historical text regarding Shakespeare. Wondering about the blocking on so-and-so's 1937 performance of Midsummer Night's Dream in the West End? They probably have so-and-so's director's notes on file. You can actually apply for programs here as well. There is, of course, a theater in Stratford, one of the Royal Shakespeare Company's venues. I think there are typically two plays rotating in Stratford at any given time. Shakespeare's home, his mother's home, his wife's home, and so on and so forth. One of them has a lovely garden you can tour and another (maybe the same one) has a renowned tea shop which is as good a place as any to have your requisite tourist tea and scone.
Other than Shakespeare, Stratford is just a beautiful town to wander around. It's got tons of small shops to check out, depending on your interests. Unlike some other historic British towns (Bath, for example), they've managed to keep McDonald's and the Gap out of their town, which is nice. Be sure to get off the beaten path a bit - there's lots to discover down side paths and back roads - just be sure to respect people's private property (they may be used to tourists, but they certainly don't want you wandering around their front porch trying to figure out if they're a museum or a private home!!) - if something is open for business they usually make the fact quite clear.
Walk by the canal and visit some of their old churches. There is, of course, a brass rubbing center if you're in to that. You can take a river boat or canal cruise if the weather's cooperative. You can learn about John Harvard - Stratford's other famous citizen - at the Harvard house. Be sure to check out the Falstaff Experience if you like campy touristy goodness. And to cap it all off, you have a Teddy Bear Museum and a Butterfly Farm.
Stratford-upon-Avon is an easy train ride day trip from London, or there are plenty of places to stay in Stratford.