So there isn't really a single "scientific method." But we can talk about experiments a bit, as one possible technique that people investigating the natural world can use. But we should be clear that experimentation is not the only way to do science, and indeed is a relative latecomer to how people we might today identify as "scientists" worked. Observation is a much earlier and more common mode of "science," for example, and in some fields, pure deduction is another mode. Both of these were far more common in the early modern period than experimentation, for example; Galileo did no "experiments" of consequence, but he made extensive use of observation (via an instrument that expanded the possibility of his senses, the telescope) and deduction (reasoning through his observations as to what they might mean, as well as doing some of the hard math).

(Complicating all of this is that the word "experiment" is itself a bit tricky linguistically. In English, we distinguish very clearly between experience and experiment but in many languages they are the same word. You can see that there is a really different implication between the two in English, and the fact that there isn't in some languages gives some hint at the variations in interpretation one could apply to the past.)

"Experiment" as we think of it today is usually credited to the early modern period, the very early "Scientific Revolution." There are no doubt examples of people doing things we might consider to be "experiments" in a loose way earlier than this. And there are, in fields like alchemy, a rich history of "try a bunch of things out and see what works." But what I mean by "experimentation" here, and what makes "experiment" a somewhat unique approach to learning knowledge, is something a bit different than that. It is about creating deliberately artificial conditions that allow the experimenter to isolate some aspect of reality and then make careful observation of it. And for us to really deem something worthy of being an "experiment," we usually want it to be systematic — not just a casual attempt, but something part of a real research program, where experiment is the center of the work.

Robert Boyle is probably not the first person to do "experiments" of this sort but he is generally credited as the scientist-type who tried to formalize the epistemology of this, which is to say, to push this as a new way to generate knowledge about the natural world. That this would be the case was not at all obvious and in fact deeply contested, despite seeming pretty normal to us today. Western European science had evolved out of a large Aristotlean tradition of natural philosophy, which emphasized observation primarily, with a goal of understanding the natural and regular patterns of the broader world. This was also deeply interlocked with an almost reverence of deductive philosophy as the mode to ultimate knowledge, which is to say, sort of making assumptions that you already somewhat understand how the world works and going forward from there.

Boyle's argument was, more or less, we really probably don't understand how the world works, we need to assimilate new information (inductive reasoning), and we can generate new information by the means of instruments that create artificial conditions. The use of instruments in general had already been taking off in Western Europe and were a key aspect of the "Scientific Revolution" — the telescope, the microscope, and in Boyle's case, the vacuum pump ("air-pump"). What Boyle was doing was creating a deeply artificial condition, something that could not arise in nature by itself — a vacuum chamber. Then he was doing experiments within that vacuum chamber to see what it told him about the nature of the vacuum and the nature of air.

I keep highlighting the artificiality of the conditions because it is part of why this was so unobvious and controversial. Again, the Aristotlean approach is about focusing on the natural and regular. Boyle is doing exactly the opposite of that; he's creating unnatural conditions, and then trying to infer knowledge about the natural and regular from those unnatural conditions. Today we would say he is isolating his variables, among other things, because you can't learn about things like air pressure very easily without such unusual conditions. Boyle definitely "won" this battle, in that his approach became adopted as part of "science" and most people have never heard of his detractors and their arguments, but it was a battle nonetheless, which is historically interesting.

What were the arguments of his detractors? Not just that "Artistotle was right." They had a few good ones in there as well. One was that most of Boyle's experiments were not replicable, because he had one of the only such instruments in Europe, and they were very hard to make. Even others who did have air-pumps found it difficult to get Boyle's results, which Boyle attributed to them not having as good air-pumps as he did (which was probably true; remember that each of these are essentially hand-made instruments, and they did not have plastics or other materials that we today would use to make good vacuum seals — at one point, Boyle was using cheese as part of his sealant material!).

If you can't replicate an experiment, how good is it? The answer depends on how much you trust Boyle. Boyle would try to increase your sense of "trust" by bringing in "reliable" observers (read: high-class Britishers) to watch his experiments, and then to attest that they indeed saw what Boyle describes. But now look at what we've had to do: we've moved from an observational model ("just go take a look, you can see it") to a trust-based model ("I did a thing, I wrote about it, you have to just believe me"). Is a trust-based model better? You can mitigate this, of course, by expanding the network of researchers, and doing replication, and standardization of instruments, but I just want to point out how much work that is when you are starting from "scratch," essentially. And if for some reason you don't trust Boyle and his friends — because you think they are all upper-class snobs with an axe to grind — then you are in something of a pickle as well, because what's your argument against them? "I don't trust you," works a little bit, but it's not really a grounds for a full-on scientific dispute, is it?

Hence Thomas Hobbes, the famous political philosopher, argued that Boyle's approach to knowledge was not only bad epistemology (you couldn't use devices that create artificial conditions to generate lasting truths), but also bad civics: it wasn't going to lead to people agreeing with each other, it was going to lead to endless argument and potentially even war. It rested too much on exclusionary circles (like the Royal Society of London), of work done in closed-off spaces (laboratories), of the results of idiosyncratic machines that are owned by only a tiny number of people (like the air-pump), of the testimony of other like-minded people (Boyle's other "expert" friends) to lead to truth. Hobbes was arguing, in a nutshell, that Boyle's approach looks like it is an "objective" approach (in that it relies on the results from a cold, dead instrument), but in reality, it is far more "social" than that, because it moves the site of knowledge from something we all have and can share (our deductive faculties) to these private spaces. Which is a really interesting argument, and some sociologists of science have actually proclaimed as the correct take (which doesn't mean that they think that we shouldn't use experiment, just that this approach to science will likely generate as much non-consensus as it does consensus, and indeed, our current atmosphere of anti-science anti-expertise seems to bear this out).

Anyway. I am not sure this is the exact answer to your question, which seems to be looking for some very, very old evidence of anything we might call experimentation. There were certainly people in the ancient world and medieval worlds who did things that we might call "experiments" if we wanted to, and in the loosest sense of "try something and see if it works," you can argue that non-human hominids certainly must have done such things, because they ended up making technologies that are by no means "genetically" determined, and would have required trial-and-error to perfect (even something as simple as a bow and arrow, which was likely invented and re-invented many times, requires that).

But in the sense that we associate it with the attempt to use artificial conditions as a means of creating new knowledge, in Europe we start to see people doing this very consciously in the early modern period. I don't know who one would list as the "first." I am sure there are medieval scholars (again, esp. the alchemist/chemist types) who were doing something like this much earlier. Ibn al-Haytham, for example, is often credited as being a pioneer in the area of experimental knowledge, centuries before Boyle (whether what he was doing would count as an "experiment" in the same way I am using the term, I am not sure, though he did use lenses and camera obscuras, which are ways of creating artificial conditions). But Boyle is usually taken as the "evangelist" of experimental methodology as a new form of epistemology, and his case highlights quite a lot of the interesting issues involved, and he is rightly seen as one of the important voices in what is (somewhat dubiously) referred to as the "Scientific Revolution" in Europe.