You have your story, you have a list of magazines, and now you have to roll up your sleeves and make sure the former arrives at the latter. Take a trip to an office-supply store and get yourself the following items page-size manila envelopes (8-1/2-by-11 inch or bigger; one per magazine on your list), letter-size envelopes (one per magazine on your list), and at least three stamps per magazine on your list.
Also make many photocopies of your story. If you're seriously invested in getting published, your expenses for a single story should run between $15 and $50.
Next, write a cover letter. Remember that your cover letter is one of thousands written by people just like you who want to get stories published, so keep it short. Editors say that just a few sentences usually suffice. If you have been published before at all, definitely let the editor know; on the other hand, you might get some sympathy points if you make it clear you're a novice willing to take any kind of constructive criticism. Throw in a line of sincere praise for the magazine that explains why you want to be published in its pages. Do not try to explain or contextualize the story. Just give your brief description of it. Don't write a lot of personal biographical information, and definitely do not beg. Make your cover letter quick and let the editor get to the story, which should speak for itself.
Now send out your story. Address it to the editor who will be reading it, by name if possible or To Whom It May Concern. Mention the name of the publication, but feel free to cut and paste in your word processor and send out 10 or 20 nearly identical letters.
Make out a pile of self-addressed, stamped envelopes (SASEs) for each one of your submission packets. Include these so the magazine can respond to you with an acceptance or rejection, and hopefully some editorial advice, without shelling out for a stamp.
Sit down and label all of your manila envelopes and collate all the copies of the story with all the cover letters and SASEs. Before you dump all of them into the mail, double-check to make sure you have an accurate list of exactly what magazines you submitted to, and when you sent out the envelopes.
Keep a detailed list of what you sent to whom and on what date. That way, you won't mistakenly deluge a single magazine with multiple submissions at the same time, and you'll also be able to track which magazines have gotten back to you and which haven't. Usually, magazines will state their response times along with their writer's guidelines; they will range from 2 weeks to 6 months.