Be persistent with publishers. Even if you have submitted a piece to a magazine and they rejected you, keep sending work to that same magazine every issue after until you get in. The worst that can happen is that the publisher will get slightly irritated at you after a year of sending your submissions every issue, and even that is rare. Publishers appreciate, generally, a writer who strives to become better and better every day.
Many great authors have received rejections for what we now call masterpieces. R.K. Narayan's Swami and Friends, an award-winning classic heralded as one of the greatest novels written in English from an author from India, was rejected by innumerable publishers. But one editor from London believed in Narayan's talent.
It only takes one person to believe in you in the publishing world. Finding that person can be a matter of chance, luck, or persistence.