A lot of people have already said this, but I think the thing I enjoyed most about reading Hamlet cover to cover was the language - specifically, recognizing famous quotations. It seemed as if every few lines I found myself saying "Is *that* where that comes from!". Of course, I knew that many of those famous quotes were from Shakespeare, but I didn't know where exactly. (So now, thanks to the group read, I do know!)
One thing made me chuckle though: in "real life", the only time I ever heard the line about "would this too, too solid flesh would melt" was in connection with weight loss (or lack thereof) so reading that brought on certain mental images Shakespeare never intended!