Even though I emphasize that there are no fairies in Russian folklore, and therefore, there can't be any "fairy tales", this is the book I use for my courses on Russian culture and folklore, and for my own reference when I need a quick English translation.
"Fairy Tales" is the older, generic designation for Folk Tales, and Russian Folk Tales are varied and often surprising. Some of them will be (almost) familiar, and good old Baba Yaga does appear in a few, but probably not in the way you'll expect.
Russian folk tales fall into three general categories: magic tales, i.e. the adventurous quests with magical helpers and objects and the obligatory happily-ever-after ending (but no fairies); the animal tales, closer to fables or cautionary tales, in which animals talk and occasionally interact with humans, and generally behave with human traits; and finally everyday anecdotes about ordinary people and ordinary lives, or soldiers returning from war; these are usually about outwitting the powerful (socially or magically) or illustrating the absurdity of human existence.
Afanasyev's collection is a classic, and still the only really good one available to the English-speaking public.