The first and more important step is visual inspection, anything broken, inflated, or more or less subtly burnt must be replaced. When you find something check the nearby circuitry good. Look for broken solder joints and resolder anything you may remotely think could fail, once on the job is very quick per additional joint.
You learn a lot observing the device working normally and the failure mode. Be prepared to quick shut down. In anything tube driven don't do it and go straightforward inside without powering on. If the device is freezed for good at least the power supply is bad.
After that check voltage regulator output good, BJT base voltage must usually be 0.7, and not much higher, mosfet gates must not short to ground.
Diode check function on multimeter inserts about 1,5V to be able to turn on small transistors and diodes.
Resistance check function on multimeter inserts about 0,3V to try to not turn on semiconductors for in circuit checking.
Capacitors are difficult ones, as you can't know they are good just for the capacitance and in circuit measurements are distorted. Usually replacing anything you suspect of with new ones is not that expensive.