Concepts of Proto-Science:
The Philosopher's Stone
This was, perhaps, the most sought-after of alchemical theories. With the philosopher's stone, the alchemist would possess the power of a god, able to progress the transformation index at will. The philosopher's stone, also called the Red stone, was never found over the three hundred years that alchemists pursued it (except in the pages of fiction).
By accelerating the natural transformations that led to the formation of the universe, the alchemist could change lead into gold, or potentially 'grow' a homunculus, a tiny man acting as assistant and familiar to the alchemist.
In addition, when ground up and boiled in water, the philosopher's stone would give rise to an Elixir of Life, which cured all ailments and made the drinker live forever.
Since the possession of the philosopher's stone would enable the alchemist to both live forever and turn base metals into gold, it is unsurprising that it was very popular. Ian Stewart and Jack Cohen wrote in 1999 that:
"The big goals of alchemy were recipes... for things like The Elixir of Life, which would make you live forever, and How To Turn Lead Into Gold, which would give you lots of money to finance your immortal lifestyle."
However, as I wrote previously, many alchemists did not prescribe to the Transformation Index theory, and although some used other methods to turn base metals into gold, many were perfectly content performing basic chemistry and some physics.
A rather less useful substance was the alkahest or universal solvent, which would dissolve anything. Despite the practical problems of storing such a solvent (what would you put it in?), many alchemists were greatly excited by the concept of the alkahest. The idea was first suggested by Arabic scholars before the Crusades, but only after the Renaissance did European alchemists start exploring the possibilities, once again without success.
While searching for the universal solvent, alchemists made stronger and stronger acids, including aqua fortis (concentrated nitric acid), and aqua regia (King's water, a mixture of nitric and hydrochloric acids) which could dissolve gold, but were left to bitterly agree that the closest thing to a universal solvent was the thing they had started out at: water.
New Chemical Methods:
While alchemists were not chemists in the modern view, they certainly developed some decent chemical skills while they were searching for their various goals.
For a start, they learned how to prepare a wide range of chemicals and materials, and modified versions of their methods are still used today. These include:
alloys (steel in particular)
acids and alkalis
alcohols other than ethanol
All of these were seen as 'progress' to the alchemists, and alchemists typically regarded any progress as beneficial to the enlightenment of mankind.
In addition, while making these innovations (and a great many more unsuccessful ones!), they discovered a wide range of modern chemical processes, for instance:
condensation & reflux
aerobic and anaerobic combustions
Distillation in particular was seen as an exciting discovery. If, for instance, rose petals were crushed and dissolved in alcohol, the distilled product (rose oil) could be made into perfume. However, what made the alchemists even more excited was their idea that this distilled oil was something they called 'essence of rose' and contained a blueprint for everything that made a rose into a rose. Alchemists worked tirelessly to obtain catalogues of natural essences, making them out of small animals as well as plants.
This was not the only misunderstanding that alchemists made while studying the world around them. Due to the fairly hazy scientific nature of their work, many mistakes were made.
For instance, alchemists claimed magnetism was a magical force which occurred because magnetic objects had magical 'sympathetic ties' between them. They used this as proof that natural magic existed, and that there were other ways of obtaining magic other than through evil diabolic means. These sympathetic ties covered medicine too. Some scholars believed the rather absurd notion that if a person was injured by a weapon, then the weapon should be treated rather than the person because of the sympathetic ties between the weapon and the wound it caused.
Less far-fetched mistakes covered optics: alchemists thought that prisms changed white light into the rainbow, rather than splitting the light as we know now; and the theory of octaves, which was believed (mistakenly) to control the motion of the planets and even defined one of the earliest periodic tables, long after alchemy had fallen in popularity.