Galileo realized, even during his earliest studies (published in his book On motion) that the speed of a falling body is independent of its weight . He argued as follows: suppose, as Aristotle did, that the manner in which a body falls does depend on it weight (or on some other quality, such as its ``fiery'' or ``earthy'' character), then, for example, a two pound rock should fall faster than a one pound rock. But if we take a two pound rock, split it in half and join the halves by a light string then one the one hand this contraption should fall as fast as a two pound rock, but on the other hand it should fall as fast as a one-pound rock (see Fig. 4.3). Since any object should have a definite speed as it falls, this argument shows that the Aristotle's assumption that the speed of falling bodies is determined by their weight is inconsistent; it is simply wrong. Two bodies released from a given height will reach the ground (in general) at different times not because they have different ``earthliness'' and ``fiery'' characteristics, but merely because they are affected by air friction differently. If the experiment is tried in vacuum any two objects when released from a given height, will reach the ground simultaneously (this was verified by the Apollo astronauts on the Moon using a feather and a wrench).
This result is peculiar to gravity, other forces do not beahve like this at all. For example, if you kick two objects (thus applying a force to them) the heavier one will move more slowly than the lighter one. In contrast, objects being affected by gravity (and starting with the same speed) will have the same speed at all times. This unique property of gravity was one of the motivations for Einstein's general theory of relativity (Chap. 7).
Also in his investigations of falling bodies Galileo determined that the acceleration of these bodies is constant. He demonstrated that an object released from a height starts with zero velocity and increases its speed with time (before him it was thought that bodies when released acquire instantaneously a velocity which remained constant but was larger the heavier the object was). Experimenting with inclined planes, and measuring a ball's positions after equal time intervals Galileo discovered the mathematical expression of the law of falling bodies: the distance increases as the square of the time.