Long before the advent of the internet and all the advancement of computer software that modern society has taken for granted, the discovery of new writing talent lay in the hands of the literary magazines. New and established writers would try to have their short stories published in magazines would try to get their stories in front of their audience while competing against each other and be paid 1¢ to 5¢ a word--a price that, per Erle Stanley Gardner's notes, was the price in the 1930s and is still the price today--if they were paid at all. Many of these authors worked on novels with hopes that the publishing firms that owned the magazines. Very rare would an unpublished author breakout with a book without readers having discovered their work in those magazines.
While some of those magazines are still around, and I would highly suggest reading them as they are still unearthing new talent and publishing legendary talent, the new world has allowed the reader to take a battering ram to the gates that were once controlled by the publishing firms. The days of electronic publishing, that has allowed anyone to create and publish anything for pennies on the dollar, has seen the rapid birth of ezines (online magazines). It started out slowly, and many ezines come and go, the ezine community that brings the new author straight to the reader has been growing since the 1990s. There are ezines for every genre and many are free to view by readers. Some of these pay writers, but most give just a publication by-line. In response to this growing trend, many of the remaining original literary magazines are available also as ezines. Some are free and some require paid subscriptions.
While it is no longer required writers breakout with short stories, the growth in the number of ezines allows writers to build up their readership and publication credits. These ezines are just a Google search away. I wish you happy reading and happy hunting for new authors and places to be published.