During the era in which Shakespeare wrote his famous plays, people in England would have spoken English, although it would sound somewhat different from the English we speak today. The definitions of many words have changed, and so, when reading his plays, it is helpful to read a version which is annotated with those changes in meaning. It's the same situation with idioms; Shakespeare and his contemporaries used many idioms that we no longer use today, and so they don't really make much sense to us.
In some ways, however, the differences in language in Shakespeare's plays have more to do with the fact that he's writing in blank verse (unrhymed iambic pentameter). Sometimes, he has to rearrange the typical order of words (this order is called syntax) so that he can achieve the meter (and sometimes the rhyme) that he desires.
Romeo and Juliet speak English, although since the language has evolved so much, some of the words or phrasing may be different than what you are used to.