There are, as you can imagine, about 1,000,000,000 websites dedicated to Shakespeare on the web. A few of them are really good, the rest of them, though they may be less then good, nonetheless testify to the enormous (indeed eternal) popularity of the plays and the playwright. Here are some of the better ones I found:
Shakespeare in Quarto - a website maintained by the British Library. It contains good background material and digital copies of the original (quarto) 17th-century texts.
The Complete Works - a fantastic resource if you're looking for a particular reference from one of the plays as the website contains its own search engine.
Shakespeare's Globe —some good background on the reconstruction of Shakespeare's Globe on the Bankside in London. A fantastic fieldtrip for anyone heading across the pond.
The Folger Shakespeare Library - the Folger is located on Capitol Hill in Washington, D.C. Another great fieldtrip if you're heading south.
Shenandoah Shakespeare Express - more information about the talented young company who will perform Measure for Measure for us on Tuesday, November 9th.
The Shakespeare Mystery - not something we talk about in class, but an abiding (some would say chronic) issue for Shakespeare hobbyists: did Shakespeare write the plays, or was it the Earl of Oxford. This is the website that accompanied a PBS Frontline series on the question of Shakespearean authorship.