Performances from Shakespeare's Globe Theatre are available online
If you don't have the cash or time to visit the recreated Globe Theatre and see Shakespeare's plays performed in a near-original setting, you'll at least have access to the next best thing. The playhouse has just launched Globe Player, a store that lets you pay to watch full Shakespearean performances anywhere in the world. You can rent videos for a week (typically £4, or about $6) if you're content to stream through the web; if you just have to keep that copy of Twelfth Night for posterity, buying the video (£8/$13) gives you a download that you can transfer to many modern devices. While the catalog is still thin at present (what, no Titus Andronicus?), there are quite a few classics to get you started. This won't compare to the full experience of seeing two hours' traffic on the stage, but it should do a good job of bringing those Elizabethan era verses to life.