Is there a comprehensive learning plan for a self-teaching beginner in data science with no programming background?
Short Answer:

Practice.  You already see it's both math and programming, so start practicing your math and your programming and eventually you'll be there.

Long Answer:

Data science happens at the intersection of math and programming.  

A lot of people start with some intense math (biotech, physics, engineering, or if you're lucky statistics) and thus have to learn the programming from scratch.

A lot of others start with being some professional developer, and then have to learn the math.

But as you said, you know neither.  So check your motivations.  Do you really want to be a data scientist, or are you just caught up in the buzz that still surrounds it?  You really have to love it to endure the massive learning curve.  And if you love it, figure out why you're only now getting started.

Alright.  Still on board?  If so, you really have three paths as I see it.


you can follow along with some mooc or series of tutorials to spoon feed you some basic project.  It is a start.
You can first try to break into being an analyst first.  Could be a much shorter path.  Best case scenario, analysts can turn directly into data scientist.  Worst case scenario, you get stuck not quite living your dream.  If you take this route with the intention of eventually becoming a data scientist, be verrry wary of the tools you use.  Highly automated analyst tools keep you locked into what the tools can do.  And if the tools don't have any statistics or machine learning baked in, then you'll get stuck.
The hard way.  If you're serious about it, then know you need to learn both overlapping languages - math and programming.  Ask around and do your research.  Find what programming languages will work best in the niche you expect you'll most like.  A part of you "do I really want to be a data scientist?" should includes the types of things you want to do as a data scientist.  Out of your answer, pull a few examples of actual projects.  Of those projects, pick the easiest and learn the math behind it.  Like financial stuff?  Perhaps some simple time series prediction on some open financial data.  Like marketing?  Perhaps start with looking up A/B tests.  Yes, this one may include also doing #1, but with the very important extra layer of sincerity around why you're doing it.  And a LOT of the job of a data scientist is being able to take a question nobody's asked yet, and figure out how to apply your tools to solve it.  Many moocs rob you of that most-necessary layer of first framing the problem before trying to solve it.