newtonian mechanics - Inertial frames

I'm just starting my study of relativity, and I have a rough understanding of the connection between inertial frames, newton's laws, and galilean transformations, but I'd probably benefit more if someone could spell out clearly what is taken as an assumption/axiom in classical mechanics (newtonian vs special relativity), and what is implied. I have a lot of loose information, and it would really help if someone could tie it all together.

I've heard that inertial frames are frames in which Newton's laws hold. Now my textbook (classical mechanics, taylor), says that Newton's first law is implied by the second, and this first law is just used to determine which frames are inertial. So if an object doesn't suddenly accelerate with the influence of a force, you're in an inertial frame. So suppose the first law holds in a particular frame. How does it follow that the second and third laws also hold in that frame?

Wikipedia says that both newtonian mechanics and special relativity assume equivalence of inertial frames. But what does "equivalent" mean in this context?

Any frame moving with constant velocity with respect to an inertial frame is also an inertial frame. I know that if frame S is inertial and observe a force F, and if the respective force F' when viewed from S' (which moves at constant velocity with respect to S), the F'=F. This is stated as "newton's second law is conserved under a galilean transformation", but I'm not sure why. When demonstrating F=F', we assume F=ma in S and F'=ma' in S', so it seems like we assume the second law is true in both frames and simply show that F=F'

Like I said, I know it's a lot of loose info, but I'd really appreciate it if someone could clarify/tie together everything

Answer

1)Definition: An inertial frame of reference is a frame of reference where Newton's first law applies (uniform motion if without external force). Now if we have other frame of references that are moving relative to this inertial frame with uniform relative velocities, then all the others are also called inertial frame of references. 2)Transformation between inertial reference frames:In Newtonian mechanics, the laws of physics are invariant under Galilean transformation. While in special relativity, the laws of physics are invariant under Lorentz transformation. The latter reduces to the former in classical limit.