I became a graphic designer without formal education. There's a lot about graphic design that they actually don't teach you in school. I did, however, earn a Bachelor's degree in an unrelated field, which certainly helped me get a job (employers almost always give preference to those with a degree when given a choice between candidates).
Being self-taught requires a lot of discipline: you need to read widely, practice daily, and be very proactive about building relationships. You also need to be constantly challenging yourself; if you're someone who is uncomfortable with criticism and feedback, this may not be a good fit.
Some things you can do prepare yourself for a graphic design career:
• Take some classes in graphic design principles and software production. There is a difference between knowing how to use the tools/software (very important), and being able to create a successful design using composition, hierarchy, etc. (fundamentally critical). Classes are helpful when you're starting out, because they can get you up to speed on the basics.
• Go online. There are a lot of fantastic resources and communities that can provide context, support, and education. AIGA, the professional association for design, is a great resource: http://www.aiga.org/guide-careerguide/
Here are a few more:
• Read about the industry, voraciously. Read books about typography, design history, composition and layout, color, the business of design, anything you can get your hands on. Read online, but read books, too (your local library is your friend). Some of my favorites for beginners and not-so-beginners:
And a few more: http://www.designer-daily.com/10-design-books-to-consider-for-graphic-design-students-18213
• Build relationships. Looks for online design communities like AIGA, and reach out to professional designers whose work you like. Ask questions. Be open-minded. Listen. You can learn a lot from working professionals. Just remember that they are often busy, so be patient. It may take some asking around until you find someone who you connect with and who can give you the attention you need.
• Finally, practice, practice, practice. Certainly start learning to use the software (Adobe Creative Suite offers free trial versions, Affinity offers free alternatives but is not widely used yet, etc). But also practice with pen and paper. Give yourself design exercises that stretch you. Share your work with others to see what impact it has on people. You will need to learn quickly how to take criticism, and how to explain your design decisions. This can be a scary and challenging process, but it's very important.
One last note on graphic design: it comes in many shapes and sizes. Print design (designing product packaging, posters, logos, books and publications, advertising, etc) is a very different beast than web design (designing web sites), or app design (designing software applications for desktop computers or mobile devices). But the fundamental principles are often the same. So start with the fundamentals. Then worry about the medium that feels right for you.
Best of luck and happy designing!