There are two main ways to publish Open Access:
There is a wide range of quality OA Journals available, but not all might fit your needs. There are also some less reputable, even predatory, Open Access journals you should avoid. Before submitting any manuscripts to an Open Access Journal, you should take into account a few considerations:
Publishing in an Open Access Journal is not the only to share your work Open Access. With the right permission from publishers, you can self-archive, and share a version of your work online (in a repository, on your personal website, or elsewhere).
For work that you are currently publishing, ask to add an Author Rights Addendum to the publishing agreement
For work that is already published, check to see whether the publisher provides any permission to publish pre-prints or post-print
Reading publisher policy information in SHERPA/Romeo:
Based on the information in the screenshot above, an author who published with the American Journal of Primatology can: self-archive pre-print, self-archive post-prints (12 months after publication) as long as certain conditions are met (such as linking to the publisher's version, only shared in certain locations, the use is non-commercial etc.).