Basic troubleshooting - Use all your senses. Sight, Touch, Hearing, Smell
Sight - does anything look out of the ordinary? Burnt component, disconnected/loose wires, bulging capacitor tops, etc?
Touch - is any component getting hot, too hot, or very hot that is not supposed to? ... - this may not be advisable for high voltage circuits, etc.
Hearing - anything buzzing, rattling, vibrating, loose?
Smell - does it smell burnt? Where is the smell coming from? Is some liquid/paste leaking from somewhere (ex: electrolytic caps)
Resistors usually do not fail... unless there was a short somewhere and then it got burnt. Easy to spot.
Caps usually do not fail... unless it's very old, or exceeded it's voltage rating, or you see gunk leaking out of it's base or top.
Active components usually are the culprit... ICs, transistors, etc. ICs are easy to replace, provided they're in sockets.
Try to determine what area of the equipment is not working. Then concentrate on that area.
If it's an audio circuit, a signal generator and scope is useful. You can probe and follow the signal... and see where the signal suddenly dropped, disappeared or went distorted. If the input to an IC chip looks good, and the output looks bad or there is no output, then most probably it's a faulty IC.
Think logically. If the equipment is dead, before you think of disassembling it, see if the equipment is plugged in!
If it's plugged in, and everything is still dead, follow the voltage... outlet, inlet, fuse, switch, transformer/psu and go from there.