Writing for your local newspaper can be a great way to break into print, and land your first paid writing assignment. I know that some people will write for free, just to develop their portfolio. I highly recommend that you never write for free - always expect some payment, even if that payment is quite small at first.
Look through the contact information for your local newspaper, and contact editors with ideas for short article ideas and or/column ideas. Please post a comment if you try this and have success!
* Query letters used to be the way you contacted any editor with a potential article idea. In a query letter, you'd introduce your idea, and why you were the perfect writer to put such an article together. Now, query letters can often be much more informal: a well-put together email is often all it takes to land an assignment. Please do research on the newspaper (or other publication) that you're querying first. To fail to do this step makes you look unprofessional, and your email will quite likely be quickly deleted. Find out the editor's name, and make sure they're the current editor. Brainstorm up some strong article ideas before contacting the editor.*