Perhaps no one else will share my opinion but I am a fan of Neukirch's approach to both local and global class field theory as presented in his book on algebraic number theory. Neukirch constructs an abstract framework including several objects and conditions which then induce a concept of a class field theory. All this is modeled upon the situation of local fields and the correspondence between prime elements and Frobenius automorphisms for unramified extensions. Hence, one has a nice motivation for this approach and moreover it is pretty elementary (although I have to admit that the verification of the multiplicativity of the reciprocity morphism is "dirty" but one can just believe this and save some time). In particular, group cohomology is not used. Neukirch then shows how to really get local class field theory from this abstract approach. The verification of the conditions mentioned above is not that hard (the existence theorem requires additional work). So, I think that this is a great path to the general concept of class field theories with local class field theory being the first example and motivation. The point is that from the same abstract framework one can also get global class field theory. This is unfortunately more technical but I think that this also provides a lot of insight.
For me the cohomological approach via Nakayama-Tate duality was always a mystery. I think if one does not learn how group cohomology appeared implicitly via the algebra theoretic considerations in class field theory, then it will remain to be a mystery. But this may just be a result of my lack of knowledge...
One could remark that a drawback of Neukirch's approach is that one does not get information about higher cohomology groups which is important elsewhere. But as Neukirch's class field axiom implies that the discrete module under consideration gives a class formation, I am not sure if this is really true...