I am curious about what is a good approach to the machinery of cohomology, especially in number-theoretic settings, but also in algebraic-geometric settings.

Do people just remember all the rules and go through the formal manipulations of the cohomology groups of class field theory mechanically, or are people actually "feeling" what is going on here. If it is the latter, could you give a list of mnemonics, or cheat-sheet, or a little fairy tale involving all the characters, so that it is easy to associate with the abstraction. Is there some strong intuition coming from algebraic-topology that would help here? Do people think of Cech covers when they do diagram chasing on crazy grids of exact sequences?

I personally was extremely happy with the central simple algebra approach to class field theory, because it gave me a whole new kind of creature, a CSA, which I could learn to adapt to and love. On the other hand, I don't see myself ever making friends with a 2-cocycle.