As for the two additional suggestions on which three plays to teach, I would highly recommend using a sonnet as an introduction to Shakespeare's style and diction. Another good option is A Midsummer Night's Dream but only for honors level or advanced readers. This play gets very complex with its enormous array of characters, which I suggest a side list for, so as to have a reference during the reading. Another buy in for this play is the sexual, bawdy banter, which is more prevalent in this than in Romeo and Juliet, but I judge my choice on what the class as a whole can handle.
I, too, would suggest Romeo and Juliet for first timers. This play can be easy for teens to identify with, due to the nature of the love story, although I normally dislike this aspect because of the potential danger of the whole suicide situation.
This is how I normally go about introducing and teaching the play. First, I have an overall group discussion regarding young love, the ages of the main characters, and what kind of decision making skills they are actually capable of. You would think teens would cheer them on, yet my experience has been the opposite. After going through the basics of what are and are not good decisions, and discussing as a group the crucial role models of the nurse and the monk, the students feel that Romeo and Juliet doom themselves from the start with their over the top, romantic, yet rash choices. My students always ask, while reading the play together and stopping throughout to discuss pivotal plot twists, etc., why is Romeo acting like such an idiot?! Another perfect discussion opportunity.
The scene with Tybalt and Mercutio is also a wonderful time in the play to talk about the risks we take for the benefit of family and friends and even our perceived enemies. Why do the events unfold as they do? Have a conversation about fate versus free will as well here, as horrible mistakes are made based upon loyalties and/or egos that directly affect the future of the main characters.
Keep in mind your audience's literacy level as well when deciding on using modernized or traditional Elizabethan language when reading the play. I teach the original text (and ultimately prefer it to any modernization) for a few different reasons. First, Shakespeare has tons of great figurative language and literary devices in its original format. These make for very successful teachable moments. Second, I enjoy teaching new students about poetic language, and go through a few samples from various plays or Shakespearean sonnets. From there, I explain how to explicate a poem, and how they can adjust their mind's filter to adjust to Elizabethan language. It really isn't that difficult once they can get used to the rhythm.
Finally, and while it may seem daunting at first, I have the students choose a favorite passage, soliloquy, etc. to memorize and recite only for me. I make time to take a student at a time, or small groups that happen to choose the same lines, to listen, give pointers on pronunciation and articulation. It builds their confidence as we read as a class. Nothing is more boring than telling students to read Shakespeare silently on their own. The comedy, dramatics, and moments for group discussion are lost otherwise. You want to give them the most rich experience you can, so they are excited and want to read even more! Trust me, it works!!
The three best and most accessible Shakespeare plays for teens are The Tragedy of Romeo and Juliet, Hamlet and Much Ado About Nothing.
While all three of the plays still use Shakespearean language, they are relatively easy to understand with most understandable and straightforward plots.
In fact, I'd argue that Romeo and Juliet is the most fun of these plays because of its raunchy language and how the impetuous actions of the teenage title characters are universal. How many teenagers don't feel like dying when their parents say they can't date someone?
All three of the plays have well-done movie adaptations as well. While the Baz Luhrmann Romeo + Juliet has its detractors, it is a fun film and has some amazing scenes. For Hamlet, students can just watch The Lion King for comparison. And I really enjoy the subtleties of Joss Whedon's Much Ado About Nothing.
Also awesome are the David Tennant (Dr. Who) stage versions of Hamlet and Much Ado About Nothing.
I truly believe that when young people experience Shakespeare, they'll realize why they should study it.