What is the procedure to follow if I want to renormalize a given operator $\cal{O}$ or a given coupling? Consider QED. I know that the renormalization constant of the mass can be obtained from considering the electron propagator, regularizing it and renormalizing it. I know that from this process we can renormalize the electron field. I know that renormalization of the photon propagator leads to the renormalization of the electric charge and the renormalization o fthe photon field.
Nonetheless, I have never seen any perspective on why we do this. I mean, is there another way to renormalize those operators and couplings? if so (as I expect) why do we use these? If I want to renormalize a given operator $\cal{O}$, is there a systematic approach of considering an specific correlation function? if I want to renormalize an specific coupling in an arbitrary Lagrangian, is there a systematic approach to follow?
I want some perspective on why we consider the things we consider when renormalizing.
 A: Each coupling and each field normalization factor potentially gets quantum corrections due to the integration over high-momentum modes
You can always consider strongly connected diagrams with appropriate external legs (e.g. two electrons for the electron normalization constant or two electrons and a photon for charge normalization).
In other words, on tree level couplings determine interaction vertices. On $n$-loop level they get corrections from strongly connected diagrams with arbitrary complex inner structure which resemble the interaction vertices (have the same types of external legs).
There is more. Because there are 4-photon (for example) strongly connected diagrams, the high-momentum modes of these diagrams can also be integrated out. This will result in the counterterm for the 4-photon coupling which is zero in the original action. The only difference is that this counterterm will be finite making this procedure unnecessary.
This is called Wilson's exact renormalization group.
P.S. In 1-loop QED you don't have to compute the vertex correction diagram. The electron charge counterterm can be computed with much less effort via the Ward identity.
