The scientific method is not really a "thing." This is one of those things that historians of science know, and people who study the actual operation of science know (anthropologists, sociologists, etc.), but everybody else is sort of gob-smacked to hear, because we use the idea of a "scientific method" as a pedagogical way to teach students how to think critically and to believe that the success of science is due to some kind of straightforward methodological principle. (This is easily the highest ratio "thing everyone in my field knows to most shock outsiders have when hearing it" thing for the History of Science as a discipline.)

But what is taught as the "scientific method" is a post-hoc rationalization that has nothing to do with how scientific work is actually done by actual scientists, and certainly is not responsible for the rise of Western (or any other) science. Even the idea that science exists or progresses primarily on the basis of "tests" is not taken that seriously by historians of science; it certainly doesn't apply to why people started believing one idea over another (there was no true test for heliocentrism until the 19th century, yet astronomers had converted to Copernicanism long before that), and there are plenty of fields of science that don't rely on falsifiable experiments to guide their work (very few ideas are truly falsifiable in a simple way, anyway).

Which is a bummer, I know, but understanding that at the beginning can open up one to actually looking into what has made science successful and what (if anything) makes it a privileged way of understanding the world.

What you are really asking, stripped of the erroneous assumption of the question, is what epistemologies did different people use? How did they decide that they "knew" something? And the answer... is going vary dramatically by the place and time. The Ancient Greeks prided abstract deduction as the ultimate form of knowledge making; you knew something if you could deduce it from sensible (in their mind) axioms. Copernicus appears driven both by a sense of aesthetics and a predilection towards sun worship. Galileo and Boyle privileged knowledge gained from experimental machines (instruments), to much controversy from others who were unsure whether such things could teach one about the natural world. Einstein privileged a sense of "completeness" and was suspicious of anything that went against this (like quantum mechanics). I pick these three very disparate examples just as a way of illustrating how different epistemological considerations can range. There are other things one could look at, such as whether they consider algebraic proof sufficient as a way to learn about physical knowledge (yes for Einstein, no for most others), whether they think experiment truly has the last word (yes for Boyle, not so firmly yes for Einstein, definitely no for the Greeks), what the goal of scientific discovery is, and so on. These things shift by time and place and person. It is only in textbooks and popularizations that these differences get smoothed out into some kind of coherent expression of the same impulse and method.

What we tend to think of as the rise of science is less about a formal epistemological method and more a set of assumptions about how the world works, what kinds of questions are going to be useful ones, and the social arrangements that allow for systematic ways of attacking these questions from many different angles. It also includes assumptions about what the point of such things are, and ideas like, "you can resolve differences by writing up your ideas and attacking other ideas, even the ideas of dead people." All of which we sort of take for granted today, but this very simple idea was not present for most of human history, and there were civilizations that had something like science that lacked this.

Anyway — there is no simple answer to this issue, because it is at the core of the serious study of the history of science, as well as the philosophy and sociology of science. But I hope I have given you a little illustration of why an obsession with a "method" is not the right answer. The idea of there being a "scientific method" is mostly a 20th century pedagogical idea, and a rhetorical weapon deployed by people who believe that scientific authority is better than other forms of authority (religious, political, whatever). One can agree with the project of such people without succumbing to the bad history that they deploy to advance it.

In terms of recommending things to read, if you are philosophically inclined, Kuhn's The Structure of Scientific Revolutions is usually what people read first on this. It is not where one's views should stop, but it is a fairly good place to start, because Kuhn frames the stakes and underlying issues very well. In terms of the history, Shapin's The Scientific Revolution does quite a lot of myth-slaying in a very concise book, and it is well-written, so it is also a good place to start.