The need of Newtons first law of dynamics
My view is similar to Fredrik's and Dale Spam's, but I'll put it a slightly different way. I see all three laws as essential. The first law is an effort to establish what we would call the inertial path. Newton didn't understand the importance of coordinate-free language, but if we extend to relativity, we can see that the First Law maps into the concept of a geodesic in spacetime-- the inertial path is the path of maximum proper time. If one does not understand the need for proper time, one simply gets Newton's version, but the point is, we still need a way to specify what the inertial paths are first, before we can say what forces do. As mentioned above, the huge revolution in Newton's way of thinking is that dynamics are about what is changing what is happening, not what is happening. (The ancients wanted motion to happen for a reason-- they would be very disappointed to discover that it is not motion, but change in motion, that happens for a reason.)
Then the second law says how forces cause deviations from the inertial path. We cannot say that F=0 means a=0 and we have the first law from the second, because we don't even know what a=0 means without the first law, that's the whole purpose of the first law. Then the third law asserts that the F that causes deviations from the inertial path must be external to the system-- it rules out internally generated forces being the cause of acceleration of a whole system. We cannot say that F=ma already tells us that if the external force is zero then a=0, because the second law refers to all forces, whether internal or external. It is the third law that says the center-of-mass is accelerated only by external forces, i.e., it establishes a meaningful difference between what is "external" and what is "internal" to a system-- forces that come in pairs are internal, forces that don't are external, that's the third law (when you make the simple extension that any system can be made arbitrarily large and thereby include all influences on that system). It's also crucial, because without a distinction between what is internal and what is external, we lose the powerful device of the "isolated system" and its conservation laws. The third law is thus tantamount to the recognition of the existence and importance of an (effectively) isolated system.