Lets take an example of something I just worked on, a dead LAMBDA LLS DC power supply with no illumination on the display. The steps I took demonstrate a process of analysis. It covers a lot of areas because the cause was exceptionally rare.
First looked for anything obvious, burnt or leaking. Then I looked to see if there was rectified line voltage, there was. Next looked for voltage on the control circuits because the display didn't light up. That would be supplied by small line transformer or small switcher chip. No power transformer so it had to be the smaller of the two switchers. 300V DC on the FET so the switching transformer was OK. It used a 2845 switcher, downloaded the data sheet to get pin outs.
I started out when almost everything was discrete. With major functions incorporated into chips, these datasheets are the quickie schematics with typical applications and voltages. Most designs don't vary much from the application notes. I sketch out a little bit of the circuit just to identify some test points like common and power. This tells me which ends of components to clip onto and gives me a place to jot down numbers. Without a paper trail you end up repeating the same tests and forgetting the numbers. It is just not possible to keep all the details in your head.
Checked gate of FET and no drive. Also checked gate resistance, bad FET almost always seems to have a shorted gate. About 8V for chip power, real iffy and about chip low voltage shutdown. Checked resistance V+ to common maybe a leaking cap. Going around the chip the voltage on the 5V REF was only 1V. That chip must be bad. I probably would have replaced it without thinking if it wasn't so hard to get to. I was still wandering about the chip power. Data sheet said nothing about 5V REF changing in a low voltage shutdown. There was also no sign of oscillation on any pin of the chip. Line switchers generally start with a little tap off of a little line voltage through a large value resistor and hen power themselves with bias winding. Those first fractions of a second at startup are a gray area in documentation.
Did a resistance check of diodes connected to that switching transformer. Besides powering itself it appeared to supply + and - voltages for op amps. Any shorted diode could dampen the output enough so the bias supply never kicked in. I even bridged in a supply cap in case the installed one had opened. I get all my LCD TV's and home entertainment equipment courtesy of manufacturers who skimp a few cents on electrolytics in power supplies. I never used to check power supply caps like I do now!
There was also a SMT op amp powered off the same V+ close by. Maybe that chip was bad and pulling down V+ power. It was used as a comparator to monitor line voltage. Why didn't they just use a two input switcher? I figured, pull one of the two inputs low and the output of the op amp will switch state. If the output flips, that will prove the op amp is still good. Pin out was the standard 2, 3 and 1. One pin had the 1V from the 5V REF of the switcher. Low but still high enough for what I was doing. The other input that I was going to bring to common was at nearly 0V already! This was connected to the 320V line power through a 562K resistor forming a voltage divider that should have produced about 8V. The output of the op amp was pulling the COMP pin of the switcher chip to common, a way to shut it down. This was to prevent the power supply for operating in low voltage conditions. Cut one lead of the resistor and made a resistance check. Replaced the failed open 562K film resistor and everything worked. Film resistor failure is very common in applications where surge pulses are common such as in series with a capacitor in a line operated device. Resistor failure is quite rare in a voltage divider with smoothed DC and conservative dissipation.
Defining a problem is a process. It is ok to make a few simple checks and follow some hunches initially. Most problems are the result of bad design causing components to operate on the edge. One can fall into a routine of going after the usual suspects. Totally random component failures require a good understanding of the circuit function. Breaking a circuit down into small sections will eventually end in results.