0: Checking the power source.
Is the power connected? Batteries charged? It can be embarrassing to report a problem when it's just an ON-OFF error. A more subtle problem can arise with batteries low on power and sometimes circuit react strangely to dying batteries. While testing, a bench power supply with known and displayed V/A/W outputs can tell you the overall circuit power usage.
1: Continuity check. Most common fault is a break in the circuit.
a) Visual check first. While checking wire connections and/or solder joints, this is also a good time to check for orientation of things like LEDs/diodes that will act like breaks in one direction. Also, check for proper pin 1 and VCC/GND connections to ICs.
b) For circuits that have been in operation for while, look for burned components like resistors and capacitors. Also, look for burned traces, and for ICs that can fail visually as well.
c) With power to the circuit off, check with DMM. Depending on complexity, you may have to test around high value resistors and ICs while still following the circuit.
2: Measuring voltage with DMM.
With the circuit powered up, start from the power input. Test the circuit at points where you know the expected values with DMM.
3: Measuring signal amplitude and form with Oscilloscope.
With the circuit powered up, start from the power input. Test the circuit at points where you know the expected values with oscilloscope.
4: Checking component values.
If nothing has been found yet, or you get measurements that are unexpected, double check component values like resistance and IC model.
5: Reading the datasheet.
Reference the datasheet for the more important components like ICs. Double check that input/output pins are correct, that V/A/W requirements are met, and unused pins on ICs are terminated correctly.
Following these steps will help you find the majority of the problems that typically plague circuits but some of them are easier said than done. When working through these, make sure that every step is accompanied by common sense and careful consideration of the results you get versus the results you expect.