Research prospective outlets either online in the classified section of Poets & Writers Magazine or by using a resource such as Novel & Short Stories Writers Market. Then check out samples of literary magazines you think might be a good fit for your work. These are available at libraries or online. You can even order a copy of a recent issue if you are unable to see one otherwise, but this expense can add up over time. But it can help you whittle down your list to only those magazines most likely to publish your work.
Find the right audience for your story by reading descriptions of magazines actively seeking submissions in the classified section of Poets & Writers Magazine. If your story has any particular ideological or cultural background, look for literary magazines that reach audiences receptive to those ideas. Some literary magazines also have special theme issues for which your story could be a good fit.
Enter a contest. You can find these at Poets & Writers Magazine along with information about entry fees, prizes and descriptions which will tell you which contests are more open to new writers.
Follow submission guidelines. These guidelines are available online at the magazine's website or through publications such as Novel & Short Stories Writers Market.
Ignore the "no simultaneous submissions" note that some magazines cite in their guidelines. "One of the most common requests made in these guideline pages is for 'No simultaneous submissions,"' according to the Noho Arts District. "Unlike the other guidelines, you can feel comfortable about ignoring this request." Chances are extremely low that your story would be accepted by two publications at the same time, and it is to your advantage to get your story out to as many outlets as possible. Still, mention that you are sending your story out as a simultaneous submission in your cover letter.