I'd like to hear your answers to this question – my own experience is doubtless atypical. Short story writers I relished as a “tween” include P.G. Wodehouse, Saki, James Thurber, Roald Dahl, Giovannino Guareschi, John Wyndham... this was 1970s England that I was growing up in... I also remember stories by Bertrand Russell and by George Mikes. And I was much taken with the mysterious short stories of Robert Graves and Graham Greene.
By my late teens, I was ready for Borges. Although come to think of it, I actually read “The Circular Ruins” long before I knew who Borges was, in a pulp anthology of horror stories. There were other amazing stories I first saw in anthologies targeted to boys, “Hope” by Villiers de l'Isle-Adam for example.
Childhood influences cannot be completely shaken off. The short story tradition is so rich, however, that there are plenty of writers working today -- writers I admire -- who haven't read many stories by the authors I just mentioned, and whose influences are quite different. This diversity is good for the field.
It was because Borges praises Kipling and Stevenson that, in my twenties, I went on to read their stories -- it's rather odd that a British writer should discover those authors courtesy of an Argentine. Chesterton's stories I'd dipped into earlier, but it was only in my twenties that I really got into him. That was also when I began seriously to read science fiction – I now believe about half of the greatest twentieth-century short stories in the English language are science fiction, but as a kid I was prejudiced against the genre. Forced to pick a best twentieth-century U.S. short story writer, I might well go with Avram Davidson.
And I think the best living short story writers in the U.S. today are Mark Helprin and Gene Wolfe. But they're both Republicans. So sue me.
Not sure I know what a "tween" is, but as a brat and a Brit in 70s England my first obsession with reading can be blamed upon The Pan Book of Horror Stories. Well packed with trash, they also introduced me to the likes of Bradbury, Faulkner, Poe, HG Wells, Bram Stoker, and less likely, Patricia Highsmith and Muriel Spark – plus, of course, Roald Dahl (whom you could not avoid in 70s Britain, nor would you want to). I think I was around ten or twelve when my mum read the back cover of issue # 9. The words, 'mother and daughter emasculate rapist' caught her eye and thereafter Pan was banned from our household, and immediately gained currency as contraband. I had no idea what 'emasculate' meant and she wouldn't tell me, hence my earliest solo venture into a dictionary. (Incidentally, Ian McEwan, Paul Theroux and Stephen King were late to the fray, and the series was already in its demise by the time they started contributing in the 80s).
My dad had me reading Biggles from an early age, projecting one of his many fixations, although, mercifully, my memory is now free of W.E. Johns' blather. Similarly, Richmal Crompton was inflicted upon me and my siblings, and that H.E. Bates fabulist. Jennings and Darbishire also played havoc with my peace of mind as a youngster, which I intend to bring up with a therapist one of these days.
The first short story to take root in my memory was Doris Lessing's 'Through The Tunnel' – it registered indelibly. John Wyndham, too – 'Consider Her Ways and Others,' being the gateway drug that got me hooked on all of his novels. In the same vein, Hugh Walters' science fiction transported me out of my dismal village in Hertfordshire, all the way to Neptune. And if I go on much longer you'll be wishing I stayed there, so here endeth the sermon.