Moscow is such a huge city with so many attractions that it would have taken a book, not just an article, to list and describe them all. What you find below is only my personal recommendations, based on my own preferences.
Igor Moiseyev’s Folk Dancing Group. This show is excellent. If you think Russia is all about classical ballet, of which you are not a big fan, try this one. It is bright, colorful, funny, and so perfectly rehearsed that you are not going to believe it.
The circus. There are two major circuses in Moscow, commonly referred to as “the old circus” and “the new circus.” I personally prefer the old one, located on Tsvetnoy boulevard (near Tsvetnoy boulevard metro station). It was founded by Yuri Nikulin, a well-known Russian actor and clown. You will see a statue of him at the entrance. The circus is always fun, I think, no matter how old you are. The program changes once in a while, but more than likely you will get to see the famous bear show with the bears riding motorcycles and bicycles. “Air gymnasts” (acrobats) are usually in the program as well, and of course magicians and clowns. Don’t forget to treat yourself to some ice cream during the break!
The zoo. You might think that a zoo is just a zoo, no matter where it is located, and you may be right. But I like animals, and I like zoos, so I recommend it. The Moscow zoo has a long history, it was opened in 1864. It has been nicely remodeled in 1990s.
The Arbat Street. The Arbat is a charming, pedestrian-only street filled with artists, musicians, street performers, and craftsmen selling their works, mostly souvenirs in the traditional Russian style. You can have an artist make a quick sketch of your face, stop for a snack at a small café, or simply walk along the street and enjoy the sights.
The Bolshoy Theater. The Bolshoy is probably Russia's most famous theater. Even those who are generally not into ballet and opera often enjoy it, and I have met people who had first discovered those arts for themselves at Bolshoy and grew to love them. Give it a try, maybe you are an opera lover as well, only you don’t know about it yet.