Hi,
Circuits should be checked using a top down approach wherever possible working from the front panel of a unit and finally down to component level. This helps to isolate a fault to a specific subsystem.
1/ Visual check for burnt components,leaking capacitors,loose components,shorted tracks etc.
2/ Voltage checks using Schematic or service manual if available.
3/ Signals can be injected and traced to narrow down a fault.
4/ When the specific suspect area is isolated then individual component checks can be done.
5/ Most checks can be done with a multimeter using the ohm meter,continuity tester and diode test feature.
a/ Resistor-Check for correct value, open or short circuit
b/ Capacitor-Check if the capacitor charges using the ohm meter.
c/ Inductor-Check for open circuit.
d/ Diode-use diode check for correct forward voltage drop (0.7 for Silicon or 0.3 for Germanium etc) and infinite reverse resistance.
e/ Transistor-Use diode check to test individual PN junctions.
f/ FET/Mosfet-Check drain source resistance.
g/ Chips-Check voltage at power pins and refer to schematic/data sheet for other voltages.
That's the fault finding method I use anyway.
Practice and persistence also helps.
HTH
David.