I may get myself into trouble here, especially because I have only been to one performance of the Bolshoi 'round the corner' from their base (too expensive, and the main building only just renovated); and have come to an unsupported opinion that the shows at the Novaya Opera are too cheap and uneven (!) But, I would strongly recommend seeing the ballet or opera at the Stanislavsky and Nemirovich-Danchenko Musical Theatre as a key part of the Moscow traveller's experience.
Why you may ask? Well because the tickets are inexpensive (under 20 pounds/$30), the theatre is small but interesting, and the performances I have seen - Stone Flower and Madame Butterfly - have been very moving. Several important provisos here. I don't know much about music, ballet or opera. I saw Madame Butterfly sung in Italian with Russian surtitles so I did not understand a thing, And it has to be admitted that the scenery had a miscellaneous quality, tending to wobble, whilst the staging asked some odd and distracting 'business' of the performers. But still I cried. Which has never happened before (to me) in an opera. Ever.
So that seems good enough reason for a recommendation.