Elizabethan drama is highly sophisticated and stylized, and is therefore more rewarding to read if you are aware of, and pay attention to, certain features. Reading these ancient plays may be compared, in this sense, to listening to a classical symphony.
For example, the first time through the first movement of such a symphony, you may choose to focus on the structure, identifying the two themes, the development section, and the recapitulation; the next time, you may listen for certain instruments, like the violins; I like to concentrate on differentiating between the often soaring instability of non-thematic material and the relief-inducing stability of the thematic material. Each experience is different, and each adds layers of understanding and depth of enjoyment.
Here, then, is a recapitulation of some of the kinds of things you can pay specific attention to as you read Elizabethan drama; you can focus on
(1) simply getting the meaning of the speeches and following the plot; this is probably what you want to focus on your first time through.
(2) engaging the iambic pentameter (you can do this by mouthing the words to yourself).
(3) picking up on the use of “you” versus “thee” as characters address each other, and thinking about the attitudes characters are expressing, both societal and personal, especially when they switch pronouns as they address each other.
(4) identifying and following the metaphors, many of which extend over multiple lines.
(5) noting whether characters are speaking in verse or in prose.
It is difficult, if not impossible, to pay attention to all of these things at once when you first start, but, as with all skills, you become more attuned with experience. Brief tutorials are available on the Elizabethan Drama 101 page of this website to help increase your understanding of these and other topics, which will greatly increase your understanding and enjoyment of these plays.