Reddit mentions of Russian Fairy Tales (The Pantheon Fairy Tale and Folklore Library)

That book is about Greek and Roman mythology, with a little section about Norse mythology at the end. It's a good book, but it doesn't have anything to do with Russian mythology.

There isn't any comparable book available for Russian mythology, because so much less has been preserved. If there is any book out there that presents Russian mythology the way Hamilton presents Greek mythology, it is relying on an enormous amount of conjecture and should be considered as a work of fiction, rather than as any kind of record of the past. I suggest you look at the wikipedia article on Slavic mythology, especially the section on sources of information. There might be books describing in detail what can and can't be known from those sources, but I'm not an expert in the area.

However, I can suggest a good book of Russian folklore: Russian Fairy Tales, from the Pantheon Fairy Tale and Folklore Library.

Something else that might be of interest is The Song of Igor's Campaign, a medieval epic that was translated into English by Nabokov.

I highly recommend any collections illustrated by Ivan Bilibin, like this Russian Fairy Tales. The stories are pretty well-known, so it should give you the introduction you're looking for, plus the art is really magnificent. There are also collections like this that have a ton of stories in them. If you want some sort of history/analysis/criticism to go with it, you could take a look at this one about the Russian folktale or this one about Baba Yaga specifically.

Took a class on Russian mythology, folk tales, and whatnot in college. Dug up the old syllabus, and found some book titles for you:

Russian Fairy Tales by Aleksander Afanasev

Russian Folk Belief by Linda Ivanits

First is a compilation of folk tales, fairy tales, and myths, collected from across Russia and most of them are pretty entertaining. Second is a more academic look at Russian folk tales.

Not sure if this is really what you are looking for, but hopefully it will point you in the direction you wanna go.

Russian Fairy Tales!

I signed up for that class because I wanted to share an elective with my zoology major friend (I was an econ major). It satisfied a Humanities credit. Not to mention, we were both interested in Russian culture for various reasons.

And HOLY SHIT that was an interesting class. We learned how the early Russian pagans adopted the names of Eastern Orthodox Christian deities and Saints and just plastered them onto their old gods (it would remind you of the Roman Catholic/Voodoo merger). And their folktales are hilarious - it's almost like reading TVTropes, where you start to see the same themes repeated over and over. There's this one character, Ivan the Fool who is always doing stupid shit and getting beat up by his brothers. There's Princess Vasilisa and Prince Ivan (different Ivan) who are always seeking after the mysterious Firebird. There's always a bitchy stepmother sending her daughter off to the dangerous witch, Baba Yaga, who lives in a hut that stands on chicken legs. There are tons of quest items buried inside of an egg, inside of a duck, inside of a rabbit, in a silver chest, underneath an oak tree (this oddly specific nonsensical situation happens A LOT, I don't know why).

If you're ever interested in reading Russian folktales, I recommend this book (it was essentially our textbook for the class). I guarantee hours of entertainment and WTF moments.

Then perhaps a book about Russian folk/fairy tales? I haven't read it myself, but it's got good reviews.