What is really sought when we purpose Einstein's postulates in Special Relativity? Special Relativity can be motivated by looking at Maxwell's Electrodynamics and noticing that there is some kind of inconsistency between it and Newtonian Mechanics. Indeed, as Einstein pointed out on his paper the major two issues are:


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*There are many electromagnetic phenomena where what really matters is the relative motion between two observers. What happens here is that if each observer tries to apply the laws of electrodynamics, each of them will arrive at different descriptions, whereas the observed phenomena will be the same. This suggests that somehow, the same frames on which the laws of mechanics hold in the same form should be the ones on which the laws of electrodynamics holds in the same form.

*The electric and magnetic fields were thought to be stresses on a specific material medium called the luminiferous aether. More than that, it was believed that the laws of electrodynamics should hold with respect to the aether frame inasmuch as light had speed $c$ relative to its propagation medium, which again would be the aether. Now, at time Einstein proposed the theory, there had been a quite reasonable number of failures in detecting this medium and its frame of reference. This would imply that there was no absolute frame to formulate the laws of electrodynamics.
Those are the main motivations of the theory. There is one quite subtle point, though, that I always wanted to understand better: when Einstein proposed the theory in his well-known article, he stated the motivation and then stated the postulates. But what when we purpose those postulates, what is really being sought?
I mean, what Einstein was really looking for was a new way to transform between reference frames that would keep the laws of electrodynamics invariant inasmuch as the Galilean transformations keep the laws of Newton invariant?
In that sense, what Einstein saw was that the principle of relativity from Galileo should work for electrodynamics, but in the form it was written it couldn't work, because there was one supposition of absolute time. In that sense, Einstein rederived the transformations based on that principle without absolute time to make it work for the laws of electrodynamics? So in the end, the first objective was to arrive at the Lorentz transformations?
 A: 
"What Einstein was really looking for was a new way to transform between reference frames that would keep the laws of electrodynamics invariant inasmuch as the Galilean transformations keep the laws of Newton invariant?"

It wasn't so much about finding the transformations, because the Lorentz transformations had been known for a while at the time, since about the 1890's if I remember correctly. It was really about understanding what was missing from the fundamental principles of Newtonian mechanics that lead to conclusions different from those of electrodynamics. If you'd like, it was an early instance of the search for the Theory of Everything.
In Einstein's time, the theoretical framework of Galilean mechanics was already well polished, its conclusions believed to be self-contained and self-consistent. The reasoning was then that if the entire Universe was abiding by the immutable laws of mechanics, then all phenomena should be consistent with those laws. Yet  electrodynamics was obviously at odds. This meant that either electrodynamics was faulty or incomplete, or something was amiss regarding the laws of mechanics. The latter, however, had not been challenged since the time of Newton, while Electrodynamics was a much more recent field. For a good a while most efforts were directed at testing possible electrodynamic loopholes and less understood consequences, but eventually it became apparent that Maxwell's eqs. were not at fault either.
Einstein's bold step at this point was to follow the next logical option, however unlikely it sounded at the moment: accept electrodynamics, amend instead the foundations of mechanics. What was needed really was a new mechanics invariant under the Lorentz transformations instead of the Galilei transformations. But the Galilei transformations follow from very fundamental kinematic principles: homogeneity and isotropy of space, and the principle of relativity. There was no reason to rebuke any of them, but there was already talk and evidence that the speed of light sets an invariant upper limit. So the questions Einstein saw as essential and set out to answer were: 


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*Can the speed of light postulate, added to the basic premises previously used to derive the Galilei transformations, justify instead the Lorentz transformations? Can it unify mechanics and electrodynamics?

*If so, how else does the speed of light postulate alter our understanding of kinematics, dynamics, and mechanics in general? 
A note is in order: 
The above is admittedly a very sketchy outline of the general line of thought, but it hopefully conveys the main idea. In addition, Einstein's contribution did not come quite out of the blue, although his groundbreaking paper contains no references to previous work whatsoever. In fact, Henry Poincaré arrived at exactly the same conclusions from very much the same premises at virtually the same time: his paper appeared only two months after Einstein's and the delay may have had something to do with the publication schedule of the chosen journal. 
