Best time to go to the Globe Theatre?

You need to get your tickets now, especially for July. The tickets go on sale around this time of year but the ticketing site is currently down for maintenance so I can't check for you to see if they are already on sale or not. We've seen several productions there, all at night. I went to see A Midsummer Night's Dream last summer and literally set my alarm for the moment they went on sale to get tickets. Groundling tickets even sell out...not as fast as seats but they do sell out. Buy them as soon as you have a date in mind.
We prefer evening shows at the Globe! It's fantastic to get out of the theater late and walk across the Millennium Bridge at night with St. Paul's all lit up on the other side of the Thames. Get to the theater early and they serve food in the forecourt. Have a Pimm's...pretend you're British! You can walk around and look at all the big names in theater that gave money to build the theater and had them stamped in the concrete pavers. Kind of like England's version of Grauman's Chinese in Hollywood.
For the show we saw, it did rain on the Groundlings but you can wear a rain poncho with a hood...you can buy a cheap one if rain is in the forecast. Don't bet on hot weather...London can be hot one day and rainy the next. Layer and be prepared for anything!
You will love London...decide in advance what you want to see and cluster your days geographically. You'll save time and save your feet! Use the tube - it's the best way to get around quickly and relatively cheaply.

There's no "quietest day" for Shakespeare plays at the Globe, pretty much every performance is sold out. I agree with cautions about the weather, remember umbrellas are not allowed as there's no space for them and they obstruct the view.

Also, remember if you get a standing ticket that you will have to stand for the whole performance except the interval. Don't spend the whole day on your feet beforehand and eat first. Almost every performance somebody passes out, and if I may be blunt it's usually young tourists who've spent all day walking around London and had one small bowl of salad for lunch because they want to keep their figures.

LOL Phillip. Brilliant remark about the young ones.

Get your tickets now. NEVER assume anything about weather in London based on your experiences in the US. Different continent. Days don't matter in an attempt to find a performance less crowded. Place is always sold out. Very popular venue as it it a wonderful place to see theatre. Saw Much Ado About Nothing there in 2011. Fabulous. Reserved seats 6 months in advance of the our May date. Select your date now and hope to get tickets. Packing light for London is easy and been discussed often on this website. Just don't know if it's easy to access the archives anymore with the new web design to read all the advice. Lastly, if you haven't purchased Rick's guidebook for London do so now. Read it cover to cover. So helpful, especially for the first timer. Fabulous city. Have great time!!!

I should say that Shakespeare plays are always sold out, but the modern plays that they do are slightly less popular and sometimes have tickets for sale on the day. (Except anything by Howard Brenton.) If it doesn't have to be Shakespeare, those are more likely to be available closer to the date.