What Is The Best Way To Enjoy A Shakespeare Play?

The first thing is to know what's going on, so it's a good idea to read a plot summary before seeing the play (you can always stop before the end if you want to keep some suspense.) Especially, make sure you know who the characters are. It's not a bad idea to read some key passages too, so that the language doesn't feel "foreign." School editions are good for this – they give help with more obscure language and draw your attention to more interesting and/or difficult parts of the play. But you shouldn't overdo it – it's not a task, just a little preparation to help your enjoyment. (An hour at the most.)
Ideally, it's best to see the play live, in a theatre or (in summer!) even outdoors – that is a fantastic experience. In a good live performance you get a real feeling of what Shakespeare was offering his audience – fast action, involvement, excitement.
Failing that, try a modern film production such as Baz Luhrmann's Romeo and Juliet, which keeps the original language while making everything clear to a contemporary audience.
Final tip: if you don't follow something, let it go and move on to the next part – there is no need to understand every word.