Well when I am looking at a circuit board, and trying to work out what is broken, I start by looking for component damage. This is sometimes physical damage, but often there is heat damage, in discoloured components or the board near the component. If the plastic packaging is distorted, that is a sure sign that the component is dead.
After that, it's a matter of trying to work out what the components do. There may be markings, and it is very common for one type of component to be used many times in one circuit. It's often useful to look on the internet for a circuit diagram of what I'm fixing. If that doesn't work, I'll draw out a circuit diagram for at least part of the circuit.
I often check the forward junction voltage drop on bipolar transistors, and that can often be done while they are in-circuit, as long as it is not powered. If the forward junction voltage drop is not in the range 0.4 - 0.7 V, the transistor is definitely broken, although for tests done in-circuit, there can be other current paths that show the voltage as being lower than it would be if the transistor were removed for testing.
Diodes can usually be checked the same way. LEDs will often light up when being tested, which is an absolute way of showing that they work.
I usually find it useful to check the resistance of resistors in-circuit. Although there are often other current paths that give a low reading, if the resistance is higher than the marked value, it is certainly blown. Also, it is rare for other current paths to give a reading many times lower than the marked value.
I hope this gives you some pointers.