Publish on the Internet. Now a lot of sites for self-expression of young authors - from the blogosphere to the virtual counterparts of samizdat journals. Register and publish your articles at one or more of them, to find its first readers. Do not get angry or despair if you will point out errors or defects. Rate how fair review and try the next time to avoid such shortcomings. Having trained "on cats", you can move on to a more serious level publications.
Select the journal where you want to be published and contact with the "expensive edition". Provide your text, in person or by mail. The ideal scenario would be to find an editor, head of the desired section to personally give him a printout and a disk with the manuscript (both must be signed and include contact information). If this is not possible, send an e-mail, but preferably an ordinary paper letter, he is less likely to get lost. Write a short accompanying text: "I, so-and-so, I offer you his story. I would be glad if you see fit to publish it on the pages of your magazine." Before sending a well-proofread the text - errors and typos do not paint a literary work.
Wait, occasionally reminding me of themselves with phone calls and/or letters. Be correct, editors are busy people. If the answer is no or it is negative, you can send the manuscript to another publication. If the journal is willing to take the story, but asks to make changes, decide for yourself whether you are ready to accept them for publication. For a beginner it is wiser to accept the edit (especially if it's small) and still see your text on the pages of the publication. Most likely, you can provide a copy that you can proudly show friends.
The shorter the story, the more likely it will be published (especially in paper editions).
If you continually deny the literary journals, try your luck in the thematic publications (e.g., journal for physicists, if you write in the science fiction genre, in a women's gloss - if stories about women's passions and so on).