It's fairly self-explanatory, once you get that the first pre-amp tube (V1) affects everything following it.
A defective V1 will make all else sound bad.
Each 12AX7 is really two tubes in one - V1A and V1B are two sides of the same tube.
Another point is the concept of cascading gain stages.
The first gain stage runs into the second, then into the third, etc., so you get more and more clipping or overdrive.
So, for example, Channels One and Two have three gain stages:
V1A, V1B, and V2A.
Channels Three and Four include the above and add two more gain stages - V2B and V3A.
It's usually the later stages that add the most distortion.
The phase inverter doesn't really affect the tone as much as the previous tubes. Simply put, its function is as an interface between the pre-amp and power section.
I'm sure some of the more knowledgeable tech-heads might have a more detailed explanation.