Language changes over time. Shakespeare's plays were written nearly 500 years ago, in a time when many people were illiterate. But, poor Shakespeare had to appeal not just to the illiterate louts who would throw fruits and vegetables if displeased with his plays. He must also appeal to his educated, wealthy benefactors who paid him for his work. Much description and bawdy puns could keep the undereducated amused along with a helping of gore and romance. The higher language and references kept his patrons happy.
Slang is one quickly changing aspect of language. I doubt you would say a movie was "the cat's pajamas" instead of the "awesome" or "wicked bad" label of today. Shakespeare used slang to appeal to the masses. Some of his slang is nearly indecipherable by students, and teachers wouldn't be allowed to explain his R rated meanings. Think of how today's adults cannot read and understand your text messages (be glad!). We study Shakespeare for a glimpse into the past and see our ancestors aren't as old-fashioned as we think. People and their motivations remain similar, only the language to describe them changes.
Anthonda49 is correct, Shakespeare used lots of slang in his writing to help make it modern and current for his audience. But you shouldn't forget that all of Shakespeare's plays are written in Modern English, the same English (or "normal" English) we use today. We don't say "Thou" and "Thee" for the singular of "You" anymore; nor do we say "whither" for "where" or "wherefore" for "why." But these words have simply fallen out of fashion. They weren't slang, but we have just discontinued using them.
Language is a dynamic changing thing, just like fashion, politics, hairstyles and almost anything else invented by human beings. Shakespeare's English, however, is Modern English just as we speak today.
An interesting thing to note, when you are feeling disconnected from Shakespeare's language, is that he made up over 1500 words that we still use today! Words like: amazement, assassination, bloody, courtship, generous, gloomy, lonely, and suspicious. Just think about how common and well-used words such as these are when you find yourself despairing that Shakespeare doesn't use "regular" English!