Determine your goals. You may want to write for personal enrichment only. Or you may want to make some pocket change or work your way into print publications and the better paying markets. Determining your goal will help you chart the best course.
Search for online publications. Writer's Digest has both an online database you can subscribe to and a book called Writer's Market that is updated annually and has listings for places that accept essays (and other types of articles) both for print publication and online publication.
Get feedback on your work. Allow other writers to read and offer feedback on your essay before submitting. Be sure to clean up punctuation and grammar to enhance your chances of acceptance.
Follow guidelines and submit your work. When you find appropriate places to submit to, carefully follow the submission guidelines on the publication's website. If the site doesn't have submission guidelines posted, email to request them.
Say thank you. Even if your essay isn't selected for publication, thank the editor for taking the time to consider your work. Editors don't get a lot of thanks, and your consideration may get you remembered favorably when you submit work in the future.
Use clips from online publications as professional experience to help you get your work into better online publications and finally better paying print publications.