Based on the experimental findings of Galileo, and probably some contemplation of his own, Newton summarized what he knows about motion in the form of three laws:
To put it simply, the first law of motion states that:
An object stays at rest, or continue with uniform velocity, unless acted upon by a resultant force.
This is the most common statement about Newton's First Law, but not necessarily the clearest in terms of physical content. The fact is, if you have not already realised, the concept of a force has not been defined yet!
Another statement of Newton's First Law, in terms of concepts already mentioned is
Inertia is the property of an object that resists change in motion of the object. Every object, whether initially at rest or moving with a certain velocity, possesses inertia. This inertia maintains the state of motion of the object (ie. at rest or moving with the same velocity) at all later times, unless an external agent acts against it.
This statement would be closer to the spirit of Galileo, but there are still undesirable elements in the statement that we can be dissatisfied with. The most glaring one would be the part about the external agent: how does the external agent act against inertia?
Here we find our earlier discussions on causes coming back into the picture: the material cause is the object of interest, the formal cause is the motion of the object of interest, but what about the efficient cause and final cause? Galileo never really did address this aspect of motion, but Newton sort of did, when he introduced the concept of a force. In this sense, we can say that Galileo studied the how? of motion, or more commonly known as kinematics while Newton also studied the why? of motion, commonly known as dynamics.
In Newton's picture of causes and effects, the external agent, whatever it maybe (your hand, the wind, an ant, gravity, et cetera et cetera), imparts influence on the object of interest by exerting a force on the object of interest. The function of the force is to effect a change in the state of motion of the object, so in this sense, forces are the efficient causes of changes in state of motion.
What about the final cause, you ask? Well, in Newton's Principia Naturalis you will find no attempt to address this. Probably such a question was never really in Newton's agenda when he seek to understand motion, and changes thereof. Effectively, through Newton's Laws of Motion, we understand motion of bodies up to the efficient cause (forces), and refrain from speaking about final causes. Philosophically, some people find such a stance objectionable, but that will not be our main concern here.
Going back to forces as the efficient cause of change in state of motion, we can argue that in fact, forces are actually the efficient causes of motion itself. This can be seen if we treat motion as something originally at rest, gaining velocity and moving, a change in the state of motion which we can associate with a force as the cause. Also, we find that after a force has acted on an object which have a certain initial velocity, it will probably be moving with a different final velocity. If no more forces act on the object, by the First Law, it will continue to move for all future times with the final velocity.
Let me put that to you concisely: before the force act, the inertia of the object fixes the state of motion to uniform motion with the initial velocity, and after the force act, the inertia of the object fixes the state of motion to uniform motion with the final velocity. Oops...! This isn't good: inertia fixing motion to two velocities? We are going to run into lots of conceptual problems if we assume this, since ideally we want inertia to uniquely fix motion when no force acts.
The way out of this conceptual dilemma is to assume that there is one inertia for each velocity, that is, an initial inertia whose magnitude depends on the initial velocity, and a final inertia whose magnitude depends on the final velocity. This then means that the force actually changes inertias! Now if that sound shocking to you, maybe I should tell you that Newton called this inertia momentum, ie. forces changes momenta. This idea, we shall see, will eventually become a statement of the Second Law.
From the crude notion of Galileo to this present idea of Newton, we can see how the concept of inertia has evolved. From his experiments, Galileo realised that fast moving objects are more difficult to slow down (and therefore they travel further up an inclined plane), while nobody will know how Newton reason it out, we can guess that he may have reasoned that an object at rest still possess an intrinsic inertia independent of motion. In our daily experiences, we would have noticed it ourselves that a heavy box is harder to be set into motion than a light book.
Newton call this intrinsic inertia the inertial mass, and an object with a larger inertial mass resists attempts to put it into motion more than an object with a smaller inertial mass. If you have so far taken inertial mass for granted when describing the motion of an object, be thankful that Newton was there in the 1700s to reason out this relation for you!
It is then Newton's genius to realize that we should define the dynamical inertia, aka momentum p as the product of the inertial mass m and velocity v.
p = m v
ie. the dynamical inertia is the product of the intrinsic inertia and kinematical inertia viz. the velocity v.
So where are we now, after understanding so much already? Moving on to
the Second Law, now that momentum is defined? No, we are still on the
First Law, because we are not done with the the First Law yet: implicit in the
First Law is the concept of an inertial frame and Galilean Principle
of Relativity.