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The classical elements consisted of earth, water, air and fire, however the "aether" element was added by Aristotle in the 3rd century BCE, which he believed made up stars, planets and other celestial objects. Aristotle was born in 384 BCE, and died in 322 BCE, likely age 61 or 62.
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Alchemy was a secretive practice based on trying to create a "philosopher's stone" which could turn any metal into gold, make people immortal and cure all disease. However what alchemy did do was show what experimentation can do, as certain recipes could create an interesting chemical reaction, which would've stunned the alchemists at the time.
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Robert Boyle was one of the first to expand upon Democritus' atomic theory, believing everything was composed of tiny indivisible "atoms", which are indestructible, in constant motion and vary in shape and size depending on the type of atom. In 1661, he published a book called "The Sceptical Chymist" where he criticized the experiments that alchemists were attempting to show salt, sulphur and mercury were the true "Principles of Things".
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Antoine Lavoisier is credit as the "father of chemistry" as they defined elements as "indivisible particles which we have found no way of separating". In their book "Elements of Chemistry" (1789) they list 33 simple substances in a table, many of which he proposed were elements. They also showed water is a compound, by forming it by combining hydrogen and oxygen in a sealed container.
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John Dalton was the first to suggest classifying elements based on their atomic weight, a revolutionary concept at the time. Dalton's atomic theory was based on the concept that every atom of a specific element was identical, but atoms of different elements would be different. He assigned atomic weights to the 20 elements he knew of at the time, which was one of the important contributing factors for the periodic table later in the century.