It depends on the device.
A stereo amplifier is a simple example. These will typically have a power supply, a preamplifier stage, and a power amplifier stage, and these sections can be tested independently to some extent. You can measure the power supply output voltages with a multimeter, and you can feed an AC signal into the input and trace its progress through the preamplifier and power amplifier stages.
Diagnosing a fault is usually more complicated than that, though. For example, the power supply voltages may be wrong because a fault in the power amplifier is overloading the power supply, and for another example, even if you find a point in the preamplifier or power amplifier where the signal disappears, you need to make other measurements (for example, DC operating voltage measurements) to find the actual cause of the problem.
A radio receiver is somewhat more complicated, because different stages operate at different frequencies and with different kinds of signals. Also, there is a kind of feedback from a later stage to an earlier stage; in general, feedback of any kind can complicate fault-finding.
Other types of equipment are much more complicated to diagnose.
For example, a smartphone or an MP3 player has a power supply, a headphone driver amplifier, and other largely self-contained sections, which can be diagnosed with standard simple techniques, but beyond them, all of the digital components are interdependent and it can be difficult to diagnose the exact fault, although the digital circuitry can still be divided into separate functional blocks. There are also many functional blocks contained within the firmware. Diagnosing faults in digital circuits is often best done by trial replacement of components; more advanced diagnosis requires advanced equipment such as a logic analyser, and intimate knowledge of the operation of the device.
Fundamentally, if you want to diagnose faults with equipment, you need (as well as the required test equipment) an understanding of how the device works internally. When you understand that, you will be able to work out how to test it.
Edit: Oh, and welcome to Electronics Point
