Jeremy Treglown praises the collected stories of Roald Dahl. I've got the volume sitting here in front of me (here's the Amazon US link) looking particularly alluring; I find myself only very infrequently really loving regular contemporary grown-up literary short stories (I love Joyce Carol Oates and Edward P. Jones; I have enjoyed recent-ish collections by, oh, who comes to mind, Nathan Englander? That's a while ago now. There must be some others... I like those Z. Z. Packer stories in the New Yorker), but both as a child and in my adult life I have had a complete passion for what might be called tales of the uncanny.
You know: not exactly science fiction or fantasy or crime but strange and exciting stories that turn up in children's anthologies and which give you as a child, then, a little taste of something richer and more rewarding than you have imagined you will find. Whose stories did I love reading as a child? Robert Louis Stevenson, the more thrilling efforts of Henry James, Saki's stories (which are funny rather than primarily uncanny, but it's the exception that proves the rule), Joan Aiken's absolutely wonderful tales and also, and most particularly, the stories of Roald Dahl. I must reread a few of these this weekend, I have a particular fondness for "Royal Jelly" (oh, and where is "The Wonderful Story of Henry Sugar"? It must be that this only collects his adult fiction, that is rather a pity, I will see if I can get the children's ones separately).
Updated: oh, and Poe! Poe, of course; that's the prototype for this kind of stuff. And Sherlock Holmes. I think that kind of story is how you get from reading children's books to grown-up ones; crime fiction also provides a useful bridge.