Hey there, Beginning Writer.

It's admirable you should want to become a freelance writer in today's market. Competition is fierce, with so many layoffs and former staffers being shifted over to vetted freelancers (see this wonderful yet depressing article in Slate). And pay rates are sad, sad, sad.

But you've determined you want to give it a shot, and who am I to dissuade you? Here's how I'd recommend beginning.
First, pick up a copy of Writer's Market. The front has useful articles about how to construct a query letter, how to target a magazine, etc. This is all assuming you want to write for magazines. If you want to write for newspapers (not something I'd recommend if you want to make money), that's a different story.

Then, get a copy of How To Write Irrestible Query Letters by Lisa Collier Cool and/or How To Write Attention-Grabbing Query & Cover Letters by John Wood. These are pretty necessary if you want to start sending ideas to magazines or publishers. They offer lots of examples and tell you the dos and don'ts.

But before you can hit up the big pubs, you'll need some writing samples. You might have already had some stuff published elsewhere: in an alumni magazine, in your community newspaper - really anything that contains your byline. In my experience, editors prefer paper-publication-based clips, but whatever you have will do. If you don't have anything, call the people you know, tell them you want to write for them, they can put you to work. You might not get paid (another sad story), but you'll be busy.

You can also get started with trade magazines. These are the publications like Concrete Monthly and Dental Instrument Fortnightly. They sound ridiculous to industry outsiders, and some of them are, frankly, but many take plain old articles that appeal to a wide audience. If you want to start out with trade mags, do the following:

E-mail the magazine asking if they assign articles to freelancers, and if so, can you mail them some samples. Also add a short graf about writing credits. You won't have any credits in that particular trade's field (unless you happen to know quite a bit about hydraulic lifts or supermarket technology), but that may be OK. At any rate, don't say expressly that you don't have any writing credits. If you can, you can express your experience in the topic the trade covers (such as, if you work in the real estate industry selling houses, you could hit up the real estate trades). Many of the magazines are desperate for good writers, so if you're a literate individual they'll be happy to take you on.

At least most of them will. Of the ones that say yes, follow up with some writing samples. Follow up two weeks later by mail.

When you get an assignment, ask them to send a few current copies of their mag so you can see their latest stuff. This way you don't have to order a copy before approaching them.

Whatever you do, try to build up bylined clips as fast as possible. Then you'll have some ammo to aim at bigger publications. I don't actually think there's anything wrong with hitting up the bigger pubs if you have no experience, as long as you have a good idea and present it well.

So, get to work, and best of luck. The moral of this game is to be dogged. You keep trying; you will be published.