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 within 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
 A: 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.
A: 
I've heard that inertial frames are frames in which Newton's laws hold.

The modern view of Newton's first law is that it defines the concept of an inertial frame. It also, at least conceptually, provides a mechanism for testing whether a frame of reference is an inertial frame. Suppose you know that no forces act on some particle. If that particle appears to be accelerating from your perspective, a reference frame that represents that perspective is not an inertial frame.
It's also important to look at Newton's first law from a historical context. Newton's first law was an extremely revolutionary statement. With that law, Newton was saying right up front that the Aristotelian physics that was still widely taught in Newton's time was fundamentally flawed.
Aristotelian physics taught that except for heavenly bodies, the natural state of some object was to come to rest. If you shoot at arrow in the air, it will eventually come to rest, embedded in the ground. If you roll a boulder down a hill, it will eventually come to rest, sitting still at the bottom of the hill. If you push a heavy object, it will move, but as soon as you stop pushing, it comes to rest. if If you throw a pebble in a pond, you'll make little waves in the pond, but eventually they too come to rest. Everything seemed to have a tendency to come to rest. That perceived natural tendency for objects to come to rest was what Newton was fighting with his first law.

How does it follow that the second and third laws also hold in that frame?

They doesn't follow, at least not logically. They are instead observational laws, the third law in particular. There are cases where Newton's third law fails. A force that arises from three (but not two) interacting bodies contradicts Newton's third law, and there are indeed three body quantum interactions that contradict Newton's third law. A force that involves some delay contradicts Newton's third law. Physicists in the latter part of the 19th century faced a serious conundrum in that Newton's mechanics and Maxwell's electrodynamics were in deep conflict with one another.
What about Newton's second law? There are (at least) a couple of different ways of looking at this. One is that Newton's second law is also definitional. It defines what "force" means. From this perspective, it's only Newton's third law that is a law of nature. This is the modern perspective of Newton's second law. It's the basis for why we write $F=ma$ as opposed to $F=kma$, where $k$ is some constant that relates force, mass, and acceleration.
In Newton's time, force was viewed as something distinct from the product of mass and acceleration. Newton devoted a good amount of space in his Principia to experiments from Galileo as tests of this law. This older point of view manifests itself in the customary units that are still used in the US. In the US, the pound is both a unit of mass and a unit of force. People in the US who use pounds mass and pounds force have to use that older style of Newton's second law, $F=kma$.
There are two ways to look at units that let us simply use $F=ma$ instead of the more verbose $F=kma$. One view is that this is merely a convenient trick. The conversion between force and mass*acceleration is still there, but subtlety hidden. The modern view is that using $F=ma$ represents something very fundamental, that force is the product of mass and acceleration. This is why the metric system's Newton is viewed as a derived unit rather than as a fundamental unit.

Wikipedia says that both newtonian mechanics and special relativity assume equivalence of inertial frames.

Acceleration is frame invariant in Newtonian mechanics. What that means is that the acceleration of a particle as observed in one inertial frame is exactly the same in all other inertial frames. It gets a bit trickier in special relativity. Removing the effects of time dilation and length contraction leads to the concept of proper acceleration, and this is frame invariant in special relativity.
A: In Newtonian Mechanics, velocity is a relative quantity. You think you are sitting still on your chair but actually you have infinite number of velocities with respect to infinite number of other objects in this universe. So velocity is not absolute. That is why while sitting still on your chair you never feel your velocity.["But what does "equivalent" mean in this context?"]. In technical language just replace "sitting still on your chair" by "while doing and experiment". Your can never know your velocity while doing an experiment in any inertial frame of reference. In other words all the laws of physics guiding your experiment remain same irrespective of your reference frame. In that sense all the inertial reference frames are equivalent. This is the meaning of Newton's first law.
But in Newtonian mechanics acceleration is an absolute quantity. Its rate of change of velocity. You always feel acceleration if you have one. Your mass is also an absolute quantity in Newtonian mechanics. So you always feel F=ma, the force. No matter in which ever inertial frame you are in, you always feel the force acting on you and if you measure that force (unlike velocity, you can measure the force acting on you) it will remain same irrespective of your reference frame. This is because acceleration does not depend on velocity which is not an absolute quantity but on "rate of change of velocity" which is an absolute quantity. This is Newton's second law.
A: An inertial frame is as you say, a frame within which the Newtonian laws hold. This selects a certain set of frames, the inertial frames, from all frames. It turns out that any inertial frames can be found from another by a velocity change. This gives a transitive action of the additive group of velocities on the set of all inertial frames. As the action is transitive, there is only a single orbit and hence a single equivalence class. Thus we say all inertial frames are equivalent. More physically, all predictions deduced from Newtons laws are exactly the same whichever inertial frame you are in, upto a velocity change.
