I know that a major goal of ancient alchemists was to transmute base metals into more noble ones, but how exactly did they go about trying to do this?
Alchemy was abandoned because it didn't work. Some of the things that did work became chemistry. No one successfully turned lead into gold. Nor did anyone successfully create the philosopher's stone. Alchemy was basically chemistry with a lot of magic thrown into it. The alchemists were sorcerer want to be's. They attempted to conjure spirits, and use spells on chemicals that they believed contained magical properties, all in a vain attempt to convert base metals into something of greater value. Even Isaac Newton, probably the greatest scientific mind of the last 500 years, accomplished nothing in alchemy, even though he experimented with it for decades.
They basically burn the metal, until it's properties revieled two parts. liquid solid. The liquid part was the elixir of life and the solid was the philosopher's stone. (this process took a long time). In other words, fire was the major substance they used in order to transmutate base metals.
I did it in school. I can't remember the chemical used, but I changed a penny's COLOR to gold by dipping it in the heated chemical using a bunson burner. It was still zinc in the center. Changing it into another element is impossible.