Correct approach for troubleshooting any computer / network issue

I'm wondering if we can come up with a set of instructions which we can follow step by step while troubleshooting any computer or network issue. I was trying to fix an issue with the screensaver on my computer and tried almost everything randomly. After I got the solution, it was quite obvious that I didn't look where it mattered the most - something that could have been eliminated provided I knew an approach to fixing computer or networking problems.

Would love to have a few inputs from computer enthusiasts. We can expect to have a healthy discussion on this and try to come up with very general set of instructions we all can follow while troubleshooting.