Newspapers are kind of starting to go the way of the dinosaurs these days, but many local papers and small papers are still thriving. One great way to get your feet wet and get some name recognition as a freelance writer, not to mention clips under your belt, is to start small. Many writers seek out opportunities as columnists in small local papers. The pay is not great, but the investment in your experience and portfolio certainly can be.
Look for small local publications and pitch a column idea to them. Even if they don't offer a great deal of pay, consider taking the job anyway. This is a great way to connect with readers and to promote future work. People who develop a fondness for a writer will be more compelled to buy your future book one day or to follow you on to bigger and better things.
Writing for a local column can also give you exposure to new contacts which can lead to more writing work for you as well. There are always benefits to connecting meaningfully with others in your community. When you work as a freelancer, connections are your bread and butter and the main source of your future income.
Another "modern" alternative to newspaper column writing can be writing for ezines (electronic magazines) with a byline in your article that links to your website. Although some ezines pay money, most do not, but will publish a byline at the end of your article helping you to gain exposure and readership. Does this mean you should routinely give your writing away for free? - Absolutely NOT. This is something you do a couple of times until you get some experience and to help you develop contacts. Routinely writing for free is eventually destructive and devalues the writing field for all of us. Be judicious.