Hey! I don't think I've talked about this before ever, so here's a set of thoughts to help you calibrate what you want from your experience.
Let's start with two practical questions.
What play are you reading?
Find the one that sounds the coolest or most interesting, the one you've always been the most intrigued by, and go for it. If it's one you recognize the name of, it's probably got enough pop culture osmosis that you won't be too at sea. The history plays are usually very focused on political concerns, the tragedies on high emotion and bloodshed, and the comedies on misunderstanding, sexy shenanigans, and the restorative power of romantic love.
If you're looking for specific recommendations, I'd go with Richard II for the history (it reads so crisp and clean you almost don't need notes), Macbeth for tragedy (it's fast and full of heavy metal shit), and Twelfth Night for the comedy (I think it has the best big longform gags).
What text are you choosing?
Shakespeare is public domain, so there's a billion free versions of all the plays. Some of these are pretty good, but they will often be transcriptions from the 19th or early 20th century printed editions. I would point you to the Folger Shakespeare Library's collection of modern edited, multi-formatted, free digital versions of the text. They have introductions that explain a lot of the editorial choices made, not to mention a lot of free historical context information about when the plays were composed on the website.
There is one drawback, and it is shared by many other online texts: a total absence of notes in the text itself, to elucidate historical references, gloss particularly strange words, or clear up confusing grammar. I think you can read Shakespeare just fine without things like this, if you can catch some of the vibes I'll talk about below, but maybe you're a person who craves the history. Craves it so much, in fact, you might actually... PURCHASE a Shakespeare book!
In which case, I recommend the printed Folger Editions, which to my mind have just the right amount of notation (and brief scene summaries!) to help someone aiming to get through the text with a more particular understanding of the language and context.
The standard for scholarly-use texts is the Arden Shakespeare series, but the editorial commentary there might overwhelm you if you're interested in a more casual experience. If you end up really intrigued by a play, however, it may be worth grabbing an Arden copy from the library and reading the introduction, which will provide a thorough overview of the play's history and how it's been approached throughout the centuries.
With that out of the way, we can get to
How to read Shakespeare for fun.
You need to remember this, you need to remember it so, so much. You must never ever forget it. Here is what you need to know in order to read Shakespeare, not just for fun, but maybe at all:
You need to remember that these are plays.
That may seem trite, but I mean it: to understand the "vibe" of Shakespeare you need to keep in mind that everything you are reading is not meant to be processed linearly, as your eyes might scan the lines of a novel while, perhaps, imagining a coherent secondary reality. No, I think Shakespeare's plays are things best understood when you keep in mind the texts are a kind of suggestion for something meant to be listened to and seen. Remember that much of the dialogue is literally poetic: part of the draw of the first audiences was to hear cool words describing funny, epic, and/or sexy shit recited out loud. But also, there are characters and plots!
Characters in Shakespeare will flatly and openly state their motivations, seeming to speak directly to each other or even the audience, and act in accordance with those statements. They will also, just as frequently if not more so, openly state their motivations, seeming to speak directly to each other or even the audience, and then act in ways that seem to contradict or undermine those statements. Thus some things that seem confusing, and which might strike you as some sort of comprehension issue, can also actually be false positives concealing the intrigue of the plot. The enjoyment of the theater is not the enjoyment of immediate understanding, but the enjoyment of watching something unfold.
Save for very sparse stage directions (note how characters often describe their own or other's actions, appearances, and dress, sometimes as cues to the other actors and often to the benefit of those in the crowd who could not see the stage as well), no character in what you're reading speaks "above" the narrative like the expository and descriptive text in a novel. Even the Chorus in Henry V might seem to have its own motivations and approaches to the material that the staged scenes might not necessarily support.
But the fact remains, especially if this is your first reading of Shakespeare: some stuff is just not going to make sense. And that's okay! It's the nature of time and art that you'll hit a few moments where you're like "What the hell is happening? What does this mean? Why are they talking about Greek mythology?"
Here is a secret, between you and me: professional Shakespeare scholars also have this happen constantly, it's how you come up with ideas for articles. The difference between the professional and the amateur is merely one of degree; most of the questions someone just starting out will have are questions someone in the history of Shakespeare studies has had, and those things become editorial notes. The reason I recommend the Folger printed editions is because they are affordable and I think their notes do the best job at answering the biggest pool of general reader "What the fuck?" questions you're apt to have if you're just starting out and reading for enjoyment.
So to say it again, because I think it is key to enjoyment here: these are plays. Keep in mind that, because their speech is poetic, Shakespearean characters are prone to speaking figuratively, and they expect that speech to fall on ears eager to unravel metaphors or bask in ornate descriptions. If you find yourself with an unglossed phrase or even a few lines you're having trouble unpacking, try reading them out loud. This can be very helpful with issues of grammar or flow. It also presents a few other opportunities: imagine the perspective of the character who delivers these lines, consider why that character would be saying these things in this context. Who are they talking to? Are they being sincere or ironic? Are there different tones they might take that change the temperature of the exchange? What makes the most sense based on the story you've gotten so far?
And keep in mind that in the end, there might not always be a single correct answer to questions like that. The nature of plays, and Shakespeare especially, is people edit the text all the time to result in performances that are brisker, cheaper, or that change the weight of interpretive points. There are many hermeneutic problems in the Shakespearean corpus--What's this guy's fucking problem? Why the hell doesn't this woman say anything for the rest of the scene?--that have no definitive answers, even with the fullest textual knowledge. The work of restaging and reinterpretation is built into the format, and you're part of stage crew now.
As an additional note, you may find helpful David Ball's very brief manual Backwards and Forwards, which walks through a method of reading plays as things-to-be-performed, using Hamlet as an example. It's very short and pairs well with reading Hamlet itself.
