Great! What a comprehensive answer.  In my experience, it's the language that appear to be difficult at first, especially as the majority of plays are written in blank verse.  This should, in principle, make the language more accessible, but often presents some confusing phrasing until you get an ear for it.

Engaging with the main themes of the play will make the play more approachable.  One reason why Shakespeare is still studied and produced dramatically is that his plays revolve around universal themes, many of which are still relevant today.  Think about Mercutio's death by stabbing, Romeo's revenge killing of Tybalt and the number of knife related deaths in London this year, many of which emerge from gang violence.

When first getting introduced to Shakespeare, the initial challenge is to be able to get your head around old English, which is often a cause of many problems for students (including me when I first started). Shakespeare makes use of many phrases and words that are archaic by today's standards, so aren't as readily understandable as they were when first written, and the format of the language can make it very challenging and difficult to understand. But this is something can easily be learned and there are many old English dictionaries that exist today. It just takes practice. 

Even when you do have a versed understanding of these, Shakespeare frequently tended to use a very subtle style of language that conveys more complex themes and deeper ideas than how they first appear and you would really have to read into it in order to extract these. Another challenge is that some plays, especially Richard III and Henry V, convey a kind of political commentary and an opinion of historical events and characters which, if you aren't very familiar with English history, can make it quite difficult to understand and appreciate. Imagine watching a film like Braveheart or Schindler's List without knowing anything about William Wallace/Oscar Schinlder and what they lived for and achieved in their lives - you wouldn't appreciate these films as much compared to having these insights. having said that, back then, few others would have had this knowledge either, so it was easier to have very biased characterisations and views about historical events - kind of like what Mel Gibson did with Braveheart. Others were also based on social themes that were very relevant for their time, which should also be understood as well. 

So, when studying a particular Shakespearean play, I would suggest reading around it first. For example, with Macbeth, which focuses heavily on the theme of witchcraft, it would be a good idea to understand how Medieval English people viewed witches to understand why Macbeth makes a great horror story. Or Romeo and Juliet, with the theme of blood feuds, which were taken very seriously in his lifetime. Twelfth Night was a romantic comedy that touched on a lot of themes around marriage, sexuality and attitudes around romance and love - kind of like a 16th century version of Love Actually. Get to know the themes of the play and understand the historical/social contexts around it, which can really help you to get a much deeper feel for the play. The language difference is something that can be easily be gotten used to with practice, but once you do get used to it, the next challenge is to read through the subtle, poetic imagery that he uses (poetic prose was popular back in the day) so that you get an idea for what Shakespeare was trying to convey through his characters and their dialogue.