AskHistorians

That depends heavily on what time period and region you mean.

In principle alchemy was a very crude natural science/chemistry of its time (hence the similar name to chemistry). When it was still called alchemy it also involved a lot of esoteric and nonscientific stuff, e.g. philosophers stone, transmuting lead to gold. During the enlightenment certain scientists wanted to make a distinction between the esoteric non-science alchemy and the proper science, hence the new name chemistry. But alchemy is the parent of chemistry and thus there are a lot of similarities.

Certain chemicals were known from antiquity (gold, copper, table salt, ethanol, ...) while others were discovered by an alchemist and then made public how to produce them. Then other alchemists tried those new chemicals in combination with already known stuff or on other new things to again create new knowledge/substance.
One big point which I am not sufficiently knowledgable about is how much esoteric stuff was just to make yourself look mysterious and not be trifled with for the common folk and how much they themselves really believed they were tempering with mystical and divine powers. As the most well known part of alchemy, the philosophers stone/magnus opum/ was a highly sought after material/procedure to create gold or silver. Obviously nothing like the existed in the past, but many alchemists tried to reach it and thereby discovered many other chemicals and procedures which then paved the way to proper science.

Since you specifically asked for Mercury, that one was obtained from cinnabar, a naturally occuring mineral with the chemical formula HgS. It is very red, so easy to identify, and occurs nearly all over the globe. Since it has a bright red colour, it easily caught the attention of humans. And at a certain point in antiquity they found out, that when you heat it up, the mineral thermally decomposes in the elements sulfur and mercury (which forms fumes). If you collect the fumes (similar to distillation) you can produce a very special liquid silver (hence its other name: quicksilver).

Another example of a chemical that was already known for a long time is aqua regia, a combination of HCl and HNO3 that is able to dissolve the king of metals, gold, hence the name. Before the pure acids could be created and combined, aqua regia was made via adding Salammoniac (NH4Cl) to nitric acid (HNO3). Salammoniac is a rare naturally occuring mineral, but alchemists created it by adding table salt (NaCl) to animal urine and feces (which are by nature rich in nitrogen containing compounds). Then this mixture was left standing for some days and the fouling created the desired compound, next to other chemicals as well. This meant that it needed to be purified. The nitric acid in turn was created by combining the solids KNO3 with CuSO4 & alum (KAl(SO4)2·12H2O), which all were known minerals, and heating this mixture.

The purification methods available to alchemists was mainly recrystallization for salts, mostly for what we would nowadays call inorganic compounds (so dissolving something and then crystallizing it by either removal of the solvent or decreasing the temperature), or distillation for liquids (mostly organic compounds). The second method involved heat, which also could lead to certain reactions happening. Thus a lot of alchemists just tried to distill EVERYTHING and see if anything new could be found in the collection bottle or in the residues that were heated.
This is also where the word "spirit" for liquor comes from: if you heated something (distillation) you released the spirit of the substance. Since hard liquor is obtained by distilling it, it was called spirit. Methanol was known as wood spirit, as it was obtained by distilling a certain wood.

The problem is that they didn't exactly have todays nuclear magnetic ressonances and mass spectrometres to analyze & characterize the newly found substances. What they did instead was testing the substance regarding its behaviour to other known substances and testing its properties (taste, smell, toxicity, ...) to see if it differs from findings of other alchemists. Then when they found something unreported (at least to them), there were two options:

Knowledge was either kept secret and only alluded to (or rather only the results were shown, but not the how to) or shared via books. Both were for the same reason: trying to make a name for yourself. Other alchemists then tried to improve on those things and publish it themselves. So already pretty similar to our modern academic system.

The new stuff they found exactly like scientists do today: by experimentation, observation and logical thinking. Although that logical thinking might not be that logical from todays point of view, as there was a long way to go to our current understanding of chemical elements, but rather talks of spirits, hidden truths and inherent mystical properties. As shown above with "king of metals" for gold and the name aqua regia for the one known substance that can "destroy" it.

I can give you two example of how alchemy was used with the original goal of creating gold, but still obtaining new knowledge:

First, in the 18th century in middle europe: Johann Friedrich Böttger, an alchemist who is often credited with inventing porcellain in Europe. He was apprentice to a chemist and then started trying to create gold from lead. After some stuff happened, he was imprisoned by August II, king of saxony, to be true to his word and create gold for the king. Obviously that didn't work out exactly and the punishment for failure would be death, making Böttger fear for his life after years of no results. A scientist, Tschirnhaus was ordered to support Böttger in his experiments and after a while could convince Böttger to instead try and create porcellain to save his neck. Böttger tested different clays from all over the kingdom of saxony and combined them in different ways and fired them in different ways. So a lot of trial and error testing. Finally he found a combination that created fine porcellain, although not yet white, that came in a later step by others. This was as good as gold at the time, since it normally had to be imported from China. Thus the original Meissen porcellain was developed and made the king very rich and thereby saving Böttgers life. The specific combination of clays and how to produce the porcellain from it was a well kept secret but as always, others found similar ways to create porcellain and went on and produced it outside of Saxony. Thus again putting alchemy in a position of secrecy and hidden knowledge.

The second example I want to provide is the Bologna Stone, which has a lot of history to it you can read in the linked articel. But the gist is the following: Vincenzo Cascariolo was an italian shoemaker by trade who tried to create gold via heating up certain minerals he found in his area. Instead he discovered a method to make a stone that could store sunlight and then emit it in darkness! He found that when he heated stones from a special location together with charcoal for a while, the cooled down stone was emitting light by itself if placed in darkness. But only for a while and then it would need to be placed in sunlight again to be recharged. How magical! Of course, nowadays we know that property under the name phosphorescence and you can buy a lot of toys for children that glow in the dark. What Cascariolo did is taking a mineral which, unbeknowing to him, contained BaSO4 and then reduced it with the charcoal to BaS, which is a phosphorescent mineral. So with a very rational & scientific procedure he could produce something that felt magical for his time.

No primary sources though, as this is all stuff I learned during my chemistry studies in university.