I imagine that Mr Tennant has a website where announcements such as, he's going to be in a play directed by so-and-so, would be announced. (And a quick Google search confirmed this.)
Once you know your travel dates, it's worth keeping an eye on the websites for the theatres, or subscribing to their e-mail lists, for season and ticket sale announcements and so forth. That way if there is something you really want to see, you won't miss out on getting tickets. For our trip this October, we bought several of our theatre tickets the day they went on sale to the public.
Theatres you might enjoy include, but are not limited to: the Globe (both the outdoor productions and the indoor ones at the Wanamaker), the National Theatre, the Almeida, Old Vic, Young Vic, Trafalgar Studios, the Barbican, and the Donmar Warehouse. There are many, many more.
I know he's going to be in Richard II at the Barbican in January, with the RSC, but that doesn't help you. Hope you are able to find something you will enjoy.
And yes, the Globe's outdoor theatre really is open-air, so dress accordingly. I'd go for the groundling (standing) ticket myself, as I don't mind standing and it's a bargain (five pounds), but if you get seats, definitely rent the cushions.