I think I must be kind of intuitive rather than logical like some of you.
I seem to start where I think there might be a problem.
If I see a broken needle I know I may have a stuck shuttle so I start there.
If I see a tension spring sticking out at an odd angle I know
I need to rebuild that tension or the machine won't work.
If it doesn't want to turn by hand, I oil it.
If there is rust, I know I need to remove it but I don't always dig in to the rusted stuff first.
I check to see if everything is all there. How deep is the rust? I have been amazed at some of the machines that were rusted and still able to be rescued.
Usually there is caked in lint - I clean that all out first. Then I oil everything.
Some times I find the problems areas along the way because the problem areas aren't all that obvious.
There are things I always clean: the bobbin area gets taken apart, cleaned and re-assembled, the tension gets taken apart, cleaned and re-assembled, the balance wheel gets taken apart, cleaned in that area and re-assembled, check the motor, etc. I don't take everything down at one time and then put it all back at one time. I do it one piece at a time. I don't even like to leave the room with things dis-assembled makes me crazy.
Some times the DGKs show up at my shop. I usually save lint cleaning for them to do. They usually fight over the cleaning brushes or the squeeze bottles - I try to help them to learn to get all of the lint out. My youngest DGD loves to clean out lint. Lint flies all over the place when she comes over. I let her use T-F a couple weeks ago and her eyes lit up big time when she was able to turn the machine. She is able to polish up with a bit of machine oil now, too. The 3 1/2-year-old, Wilbur, usually 'helps' with oiling the older machines - I don't usually let him 'help' with the newer machines unless it is hopelessly stuck. I figure they are learning and I am passing on some of what little bit I know. I can see they aren't ready for much more just yet. I have quite a few old beat up machines around. One day we'll dis-assemble a machine a bit at a time. Maybe we can make it work and maybe we will loose our screws.