Always, always start with a good visual inspection. Burned resistors, blown apart semiconductors, bulging capacitors, bad soldering, etc. Many times, a good visual is all that it takes to find the failed parts.
Capacitors can sometimes (depending upon type and usage) fail shorted, those are easy to find. Probing circuits, either with a meter or scope, can help determine if it is open. Then you remove and test either with a meter or cap tester. Resistors are either higher than spec or open. Transistors almost always fail shorted, easy to test in circuit using the diode test function with your DMM. Removing from the circuit and testing is the definitive method if in circuit measurements don’t look right. Checking the schematic for low value resistors or other paths that would allow current to flow (like a diode) can sometimes prevent a wasted effort removing a part that is okay but looked bad in-circuit.