If you ever see a play in your life, you must see a Shakespeare play in the Globe Theatre! There is absolutely no experience like it: no microphones, natural lighting, and excellent acting. The actual production was superb but what pulled it up to the highest level was the beautiful surroundings of the Globe Theatre as night fell over the Thames.
We did the tour of the Globe with an informative guide who made the theatre live for us. We were able to see the backstage people finishing off props etc getting ready for a performance that evening. The museum was very interesting as well and lots to see and learn. We came back that evening to see a performance. If you can, get a seat against the wall as it gets tiring to sit without back support. Hiring cushions makes for a better performance too and a blanket for a chilly evening. Standing up in the pit is cheap and might be authentic but I reckon a seat makes it more comfortable.
I guess the Globe theatre is well worth the visit and tour during day time also. But I can sincerely say that a play under open skies was unforgettable! I will definitely revisit if in London