It may not be enough to see that you have power - you may have to be certain you have the correct voltage coming in.
You'll need to get a schematic of that board. Passive components such as, resistors and capacitators can easily be tested.
If htey are in a series circuit their voltages should add up to the voltage coming into the series circuit. If they are in parallel then they should all have the same voltage as came into the parallel circuit.
You could also test resistance of each component, if it is outside the specified range then that component is faulty " even if it don't show signs of being burnt."
If, on the other hand, you have power coming in and no power beyond a specific point resulting in no power coming out at all. It can be assumed there is a single point that is the problem. That point may be a fuse on the board, or it could be its own power supply / power converter ( transfomer).
Test the resistance first
Test the voltages around the board at known points ( see schematic). If the resistance is within range and the voltages are within range also then it can be assumed to be good because current = Voltage / Resistance.
If two of the three quantities check out the third has to be correct, as well.
You'll need to read the bands of the resistor to find its rating and tollerance - then check and see if that is where it is. Capacitators have numbers on them to represent its value and tollerance ...Inductors are harder ... they need to be tested with an inductance meter.
The problem comes if your board uses SMT ( surface mount technology) because these components are so micro small it appears as tinnie tiny squares throughout the board. some are actually the resistors and capacitators ... so you probably will not be able to even test them... the meter probe is probably larger than most of these components
Also remember, when testing resistance of a series circuit passive component you can test directly without removing component leads.
To test resistance of a passive parallel component you must remove one lead to test component.
I would do things in an orderly fashion.
Replace any fuse
Test any transformer / onboard power supply
then :
go over the board carefully and write down the resistor number ( such as "R13), and its value & tollerance
Next come back and test its resistance to be sure that is within range on all resistors.
Next do the capacitators in the same manner.
Now go back and test voltages of those components in order .. resistors then capacitators.
Next move on to inductors ... testing voltages and resistance as stated in schematic.
Next, transistors (watch the probes [NPN or PNP transistor])
test for forward biased and reverse biased to be sure.
Any removable ICs should be removed and tested ..perhaps Radio Shack will do this for you.