Shakespeare wrote plays. Plays need to be listened to, seen and experienced. They cannot be read like a book. I teach them by having students act them out, watch film versions, and listen to them word for word on CDs. I also give them summaries before each act, so they are familiar with the plot before they try to read it.
I've tried a couple of things not mentioned above. I try to address some of his poetry before we jump into the plays. Many of his poems have words or phrases that the students have heard, and therefor we can address how Shakespeare helped develop our language. I also don't always teach each play in its entirety. Sometimes I just focus on specific sections of some, and then I hit the elements intensely. I let me students summarize sections by playing "three-headed opera star". Three students group up, and then they summarize the passage one word at a time.
When I teach Othello, we read quite a bit of it out loud. I try to set a rudimentary stage in my classroom as the students enter. They read with me directing and stopping every few minutes asking them to comment on what they just read, from the perspective of the character: "How do you feel right now?" "What made you say that?" "What answer are you hoping to get?"
I turn to the other members of the class and quiz them: "Audience, what effect does this scene have on you?" "Whom do you sympathize with?" "Why?"
The students stay engaged. Of course, we don't do the entire play--although I would if time permitted, and we have lively discussions about the decisions that the characters make and the choices Shakespeare makes as a playwright.
Students also trace a motif, such as money and jewels, throughout the play for a final paper they will write.
With Hamlet, groups of students are assigned one act. They chose a key scene from that act to present to class. Following this presentation, students provide a commentary of the scene they just performed and the act as a whole.
Nothing revolutionary, but it works for me.
You have to do what works best for you, of course, keeping in mind you're about to share something wonderful which is going to take some work for students to appreciate. I always start reading Romeo and Julietwith songs which refer to the young couple, outlining the characteristics of the pair given in the lyrics. I also ask for what people think they know about the story. Inevitably, when we make a list of those two things, we have an idealized picture of the story and its characters. We save the list, and when we're finished reading we have a good laugh at what we thought we knew or what the world wants us to remember about the story. It always works.
As the previous post stated there are tons of different ways to approach and teach Shakespeare. Let me offer a few different ways. First, you can use a comparative method. If your students know a body of literature well, then you can talk about similarites and differences. For example, if your students are sophisticated, then you can talk about how classical literature or Greek tragedy fits into all of this. Of if your students are just beginners, you can relate a play of Shakespeare to a modern remake of a movie. Another approach is to talk about themes and examine these themes from different perspectives. Finally, you could also think about putting on a play. If you did this, they would really learn it. Also they may have a lot of fun.
I am sure there are as many approaches as there are teachers. I have been studying and teaching Shakespeare for almost thirty years and, for what it is worth, this is generally my method.
For most modern readers the language is the first major problem. Flag this with students, I try asking them about words they might only have begun to use in the past few years...words like cyber, blog etc., and then explain a little about the evolving nature of language and how the words they will read in Shakespeare will be unfamiliar but that we will investigate these.
Then I try to elucidate a little on the great themes of literature. Timeless themes such as envy, love, greed..themes handled with aplomb by Shakespeare.
I look at a little history of theatre, of the world of theatre in the context of the Elizabethan period. When all the groundwork is done and I have prepared them well, I assign parts and begin a reading.
There are many excellent audio versions available if students are reluctant to read out loud and many excellent DVD version which are true to the text of Shakespeare. Many of the BBC versions are excellent in this regard.