Interesting discussion. It just so happens that I finished reading The Complete Works of William Shakespeare a few days ago. I'm still recovering from that ordeal.
Seriously, I began reading Shakespeare last summer when my computer broke down, and I was without computer for months. I thought I would finish it rather quickly, but after some 200 pages it was like hitting a stone wall. I just couldn't go on. I had to put the book down, and read something else instead.
But, stubborn as I am, I picked it up after a break, and continued to read. The secret to read such a massive work (1433 pages) is to read one play at a time, read some other book for a few days, and then return for the next play. So it took me a while, almost 14 months to be exact. While I was reading this book I read over a dozen other books. There were times when I thought I would never finish it, but I did. I'm too stubborn to quit. The reason why I bothered to read the book to begin with is because I wanted to read the classics, and Shakespeare was on my to-do list. I will never read Shakespeare again.
I mean, I might read some passages here, or some passages there (I have bookmarked some 70-80 pages), but I will never again read a full play. That doesn't mean I hate Shakespeare. I definitely don't. But I do think he's a bit overrated. The stories are, with the exception of the Tempest (and perhaps some other play) not original, but based on older works. The characters are often (but not always!) stereotypical and uninteresting. The structure in five acts is rigid and inflexible, never offering any surprises or twists. If it's a tragedy the protagonist will die, if it's a comedy the protagonist will survive, basically.
I do not like any of Shakespeare's so-called comedies. I think his true strength is in drama. It's only there we see any inkling of character depth and development. My favourite plays are Antony and Cleopatra, King Lear, and MacBeth, in that specific order.
What I do like about Shakespeare is his way with words. That is where his skills truly lies. He lets the characters burst out into profound monologues when least expected. My only problem with that is that the moment passes all too soon, and then it's back to some mundane dialogue about this or that, which I don't find very interesting. The Shakespearean monologue has a tendency to meander quite a bit.
Take Shylock in The Merchant of Venice for example. For a minute there it's a great, thought-provoking speech: "If you tickle us, do we not laugh? If you poison us, do we not die? And if you wrong us, shall we not revenge?", and so on. And after that it's back to some trivial discussion about money and stuff.
I have the same problem with the patriotic speech in King Henry The Fifth: "We few, we happy few, we band of brothers; for he to-day that sheds his blood with me shall be my brother, be he ne'er so vile"... etcetera. And then they continue to talk. Talk, talk, talk. It feels very anti-climactic, because the endings can never match these great speeches from the middle of the plays.
I guess that's the disadvantage of just reading Shakespeare instead of watching actors perform his plays. But as a reader I'm not particularly entertained. Yes, there are some memorable passages here and there, but as a whole Shakespeare's plays feel rather bland.
The reason why I prefer the three tragedies mentioned above is because I think they have a better pace than the others, and are more straight to the point, without all these meanderings.
So, to answer the question in the OP: - "What is so good about Shakespeare?" - I would say the language. Shakespeare has a way with words, an incredible vocabulary far exceeding any other writer that I know of, and he occasionally writes some very insightful thoughts about the human conditions that makes you ponder and contemplate. Like: "What's in a name? That which we call a rose by any other name would smell as sweet" (Romeo and Juliet, act two, scene two), or "It is a tale, told by an idiot, full of sound and fury, signifying nothing" (MacBeth, act five, scene five).
I believe Shakespeare is telling us something worth remembering in moments like that. It's just a pity that they are so far apart. You could easily make a summary of Shakespeare, Shakespeare's Greatest Hits, if you like, that would not exceed 50 pages. In fact, I'm planning on transcribing every bookmark I've made to a file to really get the best of Shakespeare.
Of course, everything is relative, and what I consider good might not appeal to someone else. For example, I've never been impressed with the oft quoted "Alas, poor Yorick!" speech in Hamlet. If we are talking about Hamlet I prefer: "We go to gain a little patch of ground that hath in it no profit but the name." (Hamlet, Prince of Denmark, act four, scene four). I love the cynicism such a remark provokes in me.
But like I said, such great lines are few and far between. Another line I love is Antony's words in Julius Caesar: "I come to bury Caesar, not to praise him." (Julius Ceasar, act three, scene two). Simple, and effective. I wish Shakespeare could write like that more often.
Since I'm in the middle of quoting Shakespeare I will share my absolute favourite line. It's from King Richard The Second, act five, scene five:
"I wasted time, and now doth time waste me."
Say what you will about Shakespeare, everyone has to admit that he's very quotable.
Anyway, to reiterate, I consider Shakespeare to be a genius, but only when it comes to the mastery of words. Shakespeare has obviously a very keen and mobile intellect. He shows it both in what the characters are saying as well in how they are saying it. Shakespeare has excellent rhetorical skills as well as some philosophical insights.
I don't think Shakespeare is the greatest mind who has ever lived, but I think he was better than most of his contemporaries. It does take some skill to tell the audience something about characters merely through their words. Most of the characters may be superficial constructs, but occasionally there appears a character who is willing to share his insights with the audience. I appreciate those moments.
The question whether or not Shakespeare is timeless or not is interesting. He is timeless in the sense that we are still talking about him, and his work, 400 years after his death. But Shakespeare was very much a child of his time. It's possible that he in some aspects may have been more progressive than the society he lived in (many writers are), but in other aspects his views don't go well with a modern audience. I'm talking about his blatant misogyny, of course. The Taming of the Shrew has already been discussed here, but Shakespeare's overall characterisation of women is not very positive. On top of my head I cannot think of a single female main character who doesn't complain about how weak and fragile her sex is. It gets tedious very quickly. And then we have the scene in part one of King Henry the Sixth, where Talbot is defeated in combat by a mere woman, Joan of Arc. Oh, his poor manhood never recovered from that!

Seriously, I was in my early teens when I caught The Taming of the Shrew, directed by Franco Zeffirelli, on television, and that film made a lasting impression in me. In fact, it was one of the reasons why I decided to read Shakespeare's Complete Works. I thought that the play would be as great as the film, and I was bitterly disappointed. I really enjoyed watching Elizabeth Taylor throw all kinds of objects on Richard Burton, but this, this play was just a shadow of the greatness of the film. I can summarise the play in two lines:
Petruchio: "No food for you until you behave like a proper woman!"
Katharina: "Never! (48 hours later) Well, okay, then."
Seriously, Elizabeth Taylor's Katharina put up far more of a struggle than Katharina in the book. The character development in Zeffirelli's film made perfect sense, but I didn't get it at all in the book. Maybe that's because Shakespeare doesn't get women.
I don't know. Maybe I don't get women myself. I wonder if even women understand themselves sometimes.
But my point is that Shakespeare is a misogynist. If you read his Sonnets you will notice that he is basically writing love poetry to a young man, and laments that a woman has come between them.
By our standards he's a misogynist, but for his time this attitude was probably perfectly normal.
And that is my point: he is not timeless in the sense that a modern audience can recognise themselves in the characters (I cannot), but he is timeless in the sense that we are still talking about him.
So, as you can tell, my opinion on Shakespeare is rather mixed. I don't consider him to be the greatest play writer at all. I hold Euripides as his superior when it comes to tragedy, and Aristophanes as his superior when it comes to comedies.
I want to apologise for the Shakespearean length of this post, but I really needed to get this out of my chest. Thank you for your time. Don't waste any more of it.
TL;DR Shakespeare is a kind of a genius, except when he isn't.