Let's say I have no relevant degree or experience and am a recent graduate. What steps can I take to break into the industry?
Start small. Many small newspapers are hungry for original articles and don't get enough coverage.
Report on whatever isn't being covered, even if it doesn't seem interesting to you. Once you become a regular, you're more likely to get better assignments or requests.
Start as a stringer, or remote reporter that can cover stories the paid staff can't.
Do it all- report, take photos, gather quotes and contacts.
Keep consistent and write as much as possible. Most editors I've worked with care little about degrees and want to look at previous pieces before hiring a new reporter.
You can start with entry level positions and internships that do not require journalistic experience or training; the scope of this work is often more generalized but it provides you the opportunity to gain experience and work your way up.
Alternatively, you may submit articles as a freelance writer or citizen journalist to news and magazine sites; articles have a greater chance of being accepted if they're well-written and refer to a timely event or other point of interest for your selected site's readership.
You may also decide to start up a blog where you can showcase your writing skills and attract a reader base, and at some point your reputation may lead to an invitation to join a news or media site. (I know a number of people who went from indie blogger to HuffPo columnist, Buzzfeed writer, and so on.)
And provided if you have the time, finances and motivation, you can always take a course in Journalism or get a diploma.