You need a basic understanding of how components such as diodes, transistors and op-amps function. From there you can gain insight into the signal path and begin to work your way through the circuits.
Next up, tools. A must have tool is an oscilloscope to debug AC circuits as it enables you to visualize the signal. Buy one, and learn how to use it - you don't need anything fancy for audio work. A signal generator might be helpful to enable you to generate reference or test signals but is not necessary. You can buy scopes with signal generators built in. I'd also recommend an LCR meter to measure the values of inductors and capacitors though most multi-meters have a capacitor test function (though usually limited vs a real LCR.) Its also handy to have a little bench top amp/speaker combo or a set of computer speakers for testing. Do not use headphones for testing as erratic signals and/or connections can create LOUD transients that can cause hearing damage.
I'd start with the basics as you do - check supply voltages and look for bad solder joints. If the basics check out then you need to break out the scope and start tracing the signal path. Start by probing at the signal source to make sure you do have an incoming signal and then work your way through the circuit to the output. This is where it helps to understand the basics of electronics. A schematic of the circuit you are debugging is very helpful as attempting to follow traces on a board is not fun and error prone.
Next up, tools. A must have tool is an oscilloscope to debug AC circuits as it enables you to visualize the signal. Buy one, and learn how to use it - you don't need anything fancy for audio work. A signal generator might be helpful to enable you to generate reference or test signals but is not necessary. You can buy scopes with signal generators built in. I'd also recommend an LCR meter to measure the values of inductors and capacitors though most multi-meters have a capacitor test function (though usually limited vs a real LCR.) Its also handy to have a little bench top amp/speaker combo or a set of computer speakers for testing. Do not use headphones for testing as erratic signals and/or connections can create LOUD transients that can cause hearing damage.
I'd start with the basics as you do - check supply voltages and look for bad solder joints. If the basics check out then you need to break out the scope and start tracing the signal path. Start by probing at the signal source to make sure you do have an incoming signal and then work your way through the circuit to the output. This is where it helps to understand the basics of electronics. A schematic of the circuit you are debugging is very helpful as attempting to follow traces on a board is not fun and error prone.
Some Resources: http://sound-au.com/ - can't recommend this DiY audio site enough. https://www.amazon.com/Troubleshooting-Analog-Circuits-Desig... good book I read years ago - Bob Pease was a fantastic engineer and writer.