I myself had a fair bit of struggle with English dialects both regional and in terms of time. What I found most useful as a speaker and as a linguist is actually watching different interpretations of the same plays from Shakespeare.
I would argue the following are the most influential reasons:
- Most actors receive extensive coaching on the pronunciation nuances, so it is just as good and as authentic as it can get.
- You have time to read it beforehand, as indeed you should, to prepare.
- The language is shown in context which makes the interpretation much easier.
- Watching, listening, processing it several times deepens the experience and opens up those nifty details that often escape us the first time.
If you have the opportunity, Shakespeare's Globe, live in London, is a marvelous experience. (Just sit on the side otherwise you might end up baking in the sun for 3 hours.) If you cannot be there in person, they often televise live performances to cinemas and theatres globally to audiences abroad. It is also a moving experience, I can personally attest to it. (http://onscreen.shakespearesglobe.com/find-a-venue/)
The popular Hollywood interpretations are also surprisingly good - definitely worth watching as one of several viewings (Take the 1993 version of Much Ado for one, for instance - http://www.imdb.com/title/tt0107616/). And the BBC has an incredible collection of Shakespeare (https://www.rsc.org.uk/shop/item/63083-bbc-shakespeare-collection-bbc-dvd/) available on Amazon as well.
Alternatively, Jane Austen, a hundred years later would also be an excellent choice for such an endeavour.