First I would recommend a picturebook Puzzling through exact sequences posted in the blog of 3Blue1Brown which gives a delightful visualization about exact sequences, chain complexes and so on. For me it is kind of weird that there is no discussion here about it in MSE.
I really recommend everyone to read its parts about snake lemma and zig-zag lemma, though I think there is a mistake in the part on induced homomorphism in homology. It gives some really nice intuition on what exactness really means and what really happens when we do diagram chasing.
While the picturebook is already intuitive, I wonder what can be done to further visualize and explain exactness along with the related concepts. Can we formulate a method to systematically intepret the pictures in the picturebook, thus making them rigorous? Maybe we can develop theorems that explicitly relate operations on Abel groups with set operations?
Also, I don't know whether there are other (perhaps better) visualization methods in addition to those in the picturebook. I would really appreciate it if anybody could provide some references on such methods.