ood morrow! Proper Elizabethan language is not the modern 'snooty' English of many plays and movies, nor the drawn out cockney accent; proper Elizabethan is more akin to the speech of backwood communities on the East Coast of the United States, where language has not changed significantly since the founding of those communities. Language is a living thing and evolves with time: new words are created and old ones altered. This evolution is obvious when comparing Swiss German and High German (your high school German won't help you talk to the Swiss), but even in the United States the comparison may be struck between a New York accent and New Orleans patois. Mainstream English, under the relentless influence of media's unifying force, is fast becoming a dry and brisk language, devoid of character and romance.
Altogether another reason for faire: filling that void. So, after you've put on suitable clothing to project yourself into a different era, you need conceal your speech. To this end, back to grade school! The three things to learn are: pronunciation, vocabulary, and grammar. These topics are explored in the links below.
Learning to speak passable faire Elizabethan is easy. It simply requires some practice of the pronunciation, and some memorization of common vocabulary. The problem I had when first working at faire was that I felt very self-conscious about my accent: DON'T BE! To the turkey / traveler / guest / visitor / person off the street, nearly any attempt to sound 'authentic' will be music to their ears! The workers at faire, who through practice have gotten good at speaking, aren't going to make fun of you either -- we all know it takes work and faire is the best place to practice! (Start ranting about Aye Sir and beneath my feet the earth did tilt tossing my ale hither while in a bar and you'll be lucky if you're only asked to leave...) More advice: speak slowly. Take some time and think about what you're going to say, scratch your head, look around, and finally answer.
So practice at faire or in your car or with friends (careful, this is a quick way to lose non-faire friends). You're learning a new language and you won't be perfect overnight.
Start by learning where the sounds of Elizabethan differ from modern English. This has the side effect of teaching you many short words. Then work on some other words and learn the terms of address for the different people you might meet. Finally, load yourself up on insults and words of praise, smear your face with dung and try to find a date!