What were the ideas of Galileo in relation to the nature of the motion of objects, and what experiments did Galileo perform to come to his conclusions?
To orient thinking relating to what Galileo says about the nature of motion, it may be noted that he contradicts the then authoritative ideas of Aristotle, whose ideas were accepted by the Church Fathers as inarguable dogma.
Galileo says that the action of the force of Earth upon falling objects does not relate to their mass so that a 200 hundred pound ball of lead and a 1 pound ball of lead when dropped from a fixed height will reach Earth at the same time. His experiment was to drop lead objects of unequal mass from a height of 200 cubits (legend says, from the Tower of Piza).
SAGR: But I, Simplicio, who have made the test, can assure you that a cannon ball weighing one or two hundred pounds, or even more, will not reach the ground by as much as a span ahead of a musket ball weighing only half a pound, provided both are dropped from a height of 200 cubits. (Aristotle, Two New Sciences from "Galileo's Acceleration Experiment." Michael Fowler, University of Virgina, Physics Department).
Regarding the obvious feather versus stone problem, Galileo specifies the difference of surface area and wind resistance as motion restricting factors, or opposing forces of motion (any force that retards or inhibits motion). He posited that in a vacuum, which allows for no opposing force of motion, the feather and the stone will reach the end of a calibrated fall at the same time.
Galileo posited that a ball in motion between two inclines, rolling down one incline and up the other, will lose motion and be slowed, then stopped. He suggested that the second incline acted as an opposing force of motion. This force he identified as friction existing between two surfaces. He posited that if the second incline were reduced to a zero incline, reducing friction, the ball would theoretically continue in motion until some other opposing force of motion slowed or stopped it (even a surface with zero incline generates friction with a second surface). For this, Galileo used a thought experiment (such as Einstein later used in developing relativity) that positioned a rolling ball in a figure representing the inclines, with the second incline having a secondary position at zero incline.
From this he realized the concepts of what he called inertia in relation to constant motion. Galileo posited that the natural state of matter, without the influence of opposing forces of motion (friction etc), is to be in a state of constant motion in a straight line. Inertia is the tendency to resist changes to this constant state of straight line motion: that which is in motion (unimpeded by opposing forces of motion) tends to stay in motion in one direction. This tendency is contradicted only when opposing forces of motion, like a wall, like friction, like an airplane braking speed, like air resistance, is introduced in some angle of relation to the path of motion.