How I read these books when I was a child. . . .
If you are looking for an introduction to the worlds of possibility in fairy tales, it's still a a good series. And can, of course, be read in any order since it's just collections of fairy tales. Those familiar with many tales may note some of the work done to make it a children's book -- "The Death of Koschei the Deathless" more often appears under the title "Marya Morvenva" and I think was simplified a bit here from most variants I have read.
This one tends heavily toward the French and Norwegian. Alas, in the French, it tends heavily toward Madame d'Aulnoy, who is definitely on th literary, not the folk, side. (Some of the other French ones lean toward legends with their allusions to locations and customs.) Also has others, from the Grimms, or Russian, and other places. Its version of "Twelve Dancing Princesses" was the one impressed on my memory; it was years before I met the Grimms' version. I think "The Wonderful Birch" was my favorite Cinderella tale as a child, and still is a good one.