1. Be professional. You have done the first part of your homework by visiting this site. Next, take the time to visit the individual sites of lit mags that you are interested in. Read their guidelines. For some reason, people often consider themselves exempt from rules. You're not. You must play by the rules like everyone else. It doesn't make you boring. It makes your writing accessible.
2. Familiarize yourself with a small group of literary magazines that you really like. Check them out as them out as the seasons change. This will: a) Make your writing better.
b) Give you something to say in your cover letter that makes you stand above the fray. (Editors like it when you can say, "I loved so-and-so's story “Blah-blah” in your summer issue. I have been inspired to submit a story I feel is similar...")
c) Add to your good literary karma: Giving time and maybe money to lit mags raises the chances that others will one day give time and money to what you write.
3. Get connected. You don't need to get your MFA to do this. Many of the people who get published have their MFA's for two reasons. One, getting an MFA can make you a better writer and better writers get published. Two, people enmeshed in writing programs meet writers and editors. Anyone can do this though. There are a trillion readings, conferences, cafes, parties, workshops and seminars just begging for your wonderful presence. If you go to these, you will invariably meet people who can give you advice or even a publishing leg-up. The karma thing applies here too.
4. If you go to hear an author read, stay for the Q&A. Instead of asking something useless and overly subjective like, "How many hours a day do you write?" ask this: "What was the first literary magazine you got published in?" Find out where your favorite authors made their debuts. If you love their work, your writing probably has qualities of theirs, and you might have a shot in the journals that they appeared in as young'ens.
5. Don't toss that rejection slip! Rule of thumb is that if you get a hand-written comment from an editor it means your story is close to being publishable. Editors rarely take time to comment on rejected pieces. This is a moment to seize upon! Get in touch with that editor. Ask if you can send something else. Ask if there was something specific that the staff disagreed about and if you are open to changes, ask if you could send the piece back after revising. You should be able to tell if this editor is receptive to your queries, and if so, do not take this connection for granted. Even if all it does is boost your writerly ego, it's always good to have an editor on your side.
6. Enter contests. You have to pay an entry fee generally. But sometimes this buys you a subscription to a journal. If not, consider this a donation to a good cause. The beauty of writing contests is that they are totally democratic (at least, the anonymous ones.) Your story could be read alongside a Joyce Carol Oates story, and the names would not matter. Plus, you can get a cash prize, which is likely more than you will get publishing in most journals.
7. Submit simultaneously, but thoughtfully. Make sure you know which journals publish what. Don't submit your funny story about your first pathetic attempt at sado-masochism to Isotope. And don't submit your essay about black holes to The Bellevue Literary Review.
8. Approach your writing with fierce determination. Approach publishing with a sense of humor and patience.