Where to publish research if I do not want a peer review process?
"…a quick- one, two, you're published!"
That's not how it works.
What works is:
- Make it a publicly available preprint (online repositories like arXiv, a preprint series of some institute, university, maybe via your personal website or blog…).
- Submit to some peer reviewed journal that complies with papers that are available as preprints (and there are journals that do not cost you anything; you will not get "open access" for free, but in many cases the preprint can stay freely available).
- Put the paper in your CV and add "submitted for publication".
Then the paper will be visible and checkable and it also shows that you know how scientific publishing works.
To the real question behind this, "will such non-peer reviewed papers count towards getting into a PhD program?", the answer is "probably not at all" (I'd be interested in proof that you can do worthwhile research that is regarded as such by people knowledgeable in the field, i.e., reviewers), "and it might even be harmful" (it looks an awful lot like trying to game the system, and cheating in any form is frowned upon).
You could put it on your blog, but it will probably be ignored. Part of the point of peer review is to help make sure that what you have published is new knowledge and not just a rehash of old things. So some experts in the field are asked to check. There are other reasons for peer review, but those might not be relevant here. Why do you want to avoid it?