Playbill shows the current and future listings for London and includes a link to buy tickets from the official ticket seller.
As others have noted, Theatremonkey is a great site to use, especially to check out how good the seats are. It’s better to use once you have decided on a play to see. You can enter the name of the theatre to get a floor plan that evaluates the quality of the seats and which have restricted views, etc.
London’s Time Out lists the plays in alphabetical order showing if they are drama, musicals, etc. It offers reviews as well.
Here are the critics’ top choices:
This allows you to search for plays using several criteria:
If you want to see something you can’t see in the U.S., you might look at Ian McKellen On Stage: With Tolkien, Shakespeare, Others and YOU at the Harold Pinter Theatre which is a one man show (not really a play), but a rare chance to see a legend in a small space (I saw him in King Lear last year).
There are also Christmas season pantomimes (not everyone’s cup of tea, but something different).
As for plays that I’ve seen in London, Hamilton was amazing. You may have seen the musical in the U.S, but it was easier for me to get to see it in London.
The Play That Goes Wrong (a comedy) was just pure fun and hilarious. It has all of the stereotypes of an old fashioned English murder mystery (setting, costumes, accents, etc.), but then everything does go wrong in great comedic fashion.
I haven’t seen any of the current serious dramas, so I can’t comment about them.
All the best,