What was the experimental setup and/or reasoning that Amedeo Avogadro, in 1811, used to arrive at the conclusion that "equal volumes of all gases, at the same temperature and pressure, have the same number of molecules"?
This law bears Avogadro’s name because he implicitly provided circumstantial evidence that it is true, by successfully applying it as explained below. But, strictly speaking, Avogadro’s law has not been discovered neither by experiments nor by reasoning since it had already been in the air if one had accepted the permanent chaotic movement of particles in gases as suggested by Daniel Bernoulli in 1728.
Also Dalton reflected about the equal volumes−equal numbers hypothesis before Avogadro, but discarded it as impracticable because he did not factor in the idea that gaseous elements might be polyatomic molecules; in consequence he thought that the equal volumes−equal numbers hypothesis were not useful to explain the known physical and chemical facts about gases. (According to Avogadro, Dalton had additionally some strange caloric objections against this view, see p. 59 of the link in the next but one paragraph.)
Hence, the idea that elements might be polyatomic molecules was not accepted at that time (1811). It was even declined thereafter for a long time (by Berzelius who dominated chemistry then).
Being a physicist and mathematician, Avogadro did not himself accomplish chemical reactions with gases, but he could built on existing gas density measurements and the following two results of Gay-Lussac (1808), which he discussed in his article in the Journale de physique, de chimie …, 1811 (all the following page numbers refer to this text) with the equal volumes−equal numbers hypothesis as starting point (!):
Gay-Lussac's combining law: “Not only do gases combine in very simple proportions” by volume, “but the apparent contraction of volume, which they experience on combination, has also a simple relation to the volume of the gases, or at least to that of one of them” (p. 58), and
his experimental combining volume ratios for different gas reaction, e.g. of the formation of water (1 + 1 = 2), ammonia (1 + 3 = 2) from the elements (p. 60).
When comparing all these data, Avogadro proceeded in two steps:
he discussed that, as a consequence of the equal volumes−equal numbers hypothesis, one could easily bring into line the densities of elementary gases with the relative molecular weights of the gases; for instance the density of oxygen being about 15.074 times larger than that of hydrogen (p. 59).
Then he had (p. 60) the idea that gaseous elements could consist of polyatomic molecules (“molecules composées”). Subsequently he discussed extensively the known combining volume experiments of gases in this light and showed that so all hitherto contradictions thereof could be solved.
In this way, Avogadro had simultaneously also proved by circumstantial evidence that the equal volumes−equal numbers hypothesis, from which he started his eventually successful reflections, must be true because he had demonstrated that it is one of the important factors for the causal explanation of all those gas data by particle theory.
Hence, the so called “discovery” of Avogadro's law consists merely in the success to bring all these gas data into accord, which vice versa substantiates that it is a law, whereas before this application, it was unclear how the equal volumes−equal numbers hypothesis could be generally made use of.