We have no idea! As with all old things concerning magic that stretch back more than a thousand ears, we have no idea! But we do have an estimation: the Middle Ages! Yes, the period of time that stretches back a thousand years! In the book Magical Brews (original title in Greek), an unknown wizard states that he stumbled upon a magical liquid during the preparation of a healing paste of many leaves, and some water fell into his wooden bowl from above. He stated that a plant he mixed a leaf of a plant that reacts negatively with water. But, he observed that the water didn't evaporate to vapour as was the normal effect but turned purple. He predicted that it was due to the fact that it was a mixture of many leaves, and continued to give it to his subordinate, causing warts to pop up on his face. Now, if that isn't the most dangerous prank in history!

As stated in the answer above, potions' inventions were, most of the time, an accident. There were some researchers in the magical community even then, who consulted their papers on the various properties of plants and animal parts and tried to experiment with them themselves. The idea of mixing water with it stemmed from the notion that water might make the solid mixture more tolerable, diluting its nasty taste. It has been observed that many crude ground pastes of the Middle Ages, when mixed with water, caused reactions that spawned swellings on people, such as boils and warts. There are several properties of many ingredients today, and they were all found over the span of two thousand years. Many mundane ingredients created no reaction, while the magical ones created smoke and some other things. Thus, the most progress was made with magical substances.

Some properties of many mundane and magical plants alike wizards and witches knew. The first method used was writing the properties on parchments; they would consider two plants with opposing properties, cancel out some and add some, and then proceed with the practical thing. Mixing two or three ingredients most times generated the same result as on parchment. The correct amounts of ingredients to give in were discovered much later when the uses of runes came were discovered. Nevertheless, the labour of the early Middle Ages researchers hatched the movement for the progress of magical brews.

With the help of runes. Runes were a pioneering method, which was discovered long ages back by a magical tribe in the Middle East. Runes contained magical properties that, when written or engraved correctly, opened a stash of opportunities. The runes involved in the use of finding the appropriate amount of ingredients are known as Potioneer Runes. We won't go into the process of finding the correct runes, or how the researchers of the mid-Middle Ages used them. Nonetheless, the usage of runes was prevalent in the mid and late-Middle Ages, though it has deteriorated now. They made a basin that had runes engraved in it, and after performing some spells, they molded water into shapes of runes that asked the question (in a very specific way), and the water shaped itself into the runes which answered the question. The answering capability of runes was limited, and thus they could not answer deep and complex questions.

The usage of plants in potions was known extensively in the mid-Middle Ages, but not animals. While there did exist magical creatures, there were too few known as witches, wizards, and wixes were too afraid to venture into forests or their habitats to search for them. Of course, this was the case with benign-intentioned folk, the Dark ones experimented with the animals. Dark wizards were known to be unhesitating to perform spells that would be detrimental to plants and animals, and they made the most progress within potions. That is why many potioneers consult potion books written by Dark wizards as they were saturated with information. Small animals were deemed insignificant by good folk, but small and large animals alike were experimented with by Dark wizards. While I do not support Dark wizards, without their contribution in whatever way, I don't think we would've been able to come as far as we have.