Let's say I'm improving my pistols skill from 7 to 8.
The table says I got to spend 8 weeks training.
I go to Buffalo Bill Cody who has a Charisma of 6 and an Instruction of 3.
Does Cody roll once (and assuming avg roll) reduce it to 7 weeks, 4days?
Or does he roll once each day and on avg reduce the training by three days so that I'm finished in 2 weeks?

And, yes, I know, page 105: If the player is able to find an instructor to train him and help him perfect his techniques, the time it takes improve his skill is reduced by 25 percent (round down).
But this and the rules under the instruction skill don't actually mesh and assumes every instructor is equally skilled. I'd rather have instructor's who vary, just as in real life.

I'd go with daily rolls for the Instructor but apply the 25% decrease as a limit to their overall benefit.

Also, have the character pay the Instructor for the full course of study up-front.

The instruction skill is rolled once. I don't know where you are getting the idea that you roll it once a day. The only time you would roll it more than once would be if the person is improving a skill by more than one point. For example, if someone was improving a skill from 6 to 7, then from 7 to 8, the instructor would roll to reduce both instruction times. The simpler 25% reduction rule is not compatible at all with the rules for using the instruction skill. I would use one or the other.

I was extrapolating that even 1 week is not going to be done contiguously/all at the same time.

Also PraetorGradivus, Buffalo Bill wouldn't be able to assist your character because his Instruction is only 3 so at best he can train beginner shootists with Pistols 1-3.

I'm not getting the idea from anywhere. I asked a question. Like many rules in SR5, Instruction is contradictory and/or vague. In this case, the rules under character advancement and the rules in the skills section were evidently written by two different people and the editor saw no reason to make them work together.

Climbing, for example, gives you a result. If that result isn't as far a you need to climb, clearly at the end of that action you can then try again and climb farther.
So having gotten a result on Instruction, let's say 3 hits and thus reducing training time by three days: Is it like climbing and the instructor can try again? If so, the next day, the next week, the next month?
Or is it one and done? Who knows, it doesn't say. And I for one am not going to pretend to know what the rulesmaker (the one who wrote the page 141 rules that is) intended.
Ergo, I ask the opinion of people on this forum.
Clearly, the easiest way is to just use the 25% reduction listed in character advancement.
However, that would seem to make the positive quality Sensei a bit verbose since the Instruction Pool [Social Limit] given for the Sensei has no impact on the rules in character advancement.

If I use an Instructor to improve my Instruction skill,
what would the "training" look like?

possibly use you as an assistant in training someone else in another skill.

or a mentorship, where you do the training on a skill you know, and the "Instructor" sits back and listens
while you tell someone else to "Paint the fence, Wax on, wax off!"

We do something like that in our 'Train the Trainer' program.

Basically giving them a firm grounding in the material, plus a going over of any FAQ / troubleshooting type questions that their students will invariably come up with.

Participate in the training of someone else. One of the best things you can take away from training in team sports is sharing what you can do in a way useful for others.

A good trainer will be able to help obviously, but the key skill is observation of the pupil, identifying key strengths and weaknesses, and also motivation. So an "outside" position allows you to observe. If you yourself are learning or teaching, you are preoccupied.

A good list of answers to FAQ will surely help.