I am a 30-year electronic technician. My advice, be careful. When working on power supplies you are working with input voltage that can be lethal. Ok since you asked: The two most important tools you need to check to see if something is "fried" is your nose and your eyes. A visual inspection can tell you alot. Lets start with the power supply. If nothing is obvious(no burnt smells or burnt pieces) you will need to test output voltages.To test a power supply you will need a voltmeter(multimeter) to test voltages. I do not suggest you open the power supply up and check it internally. Concentrate on the output voltages which under normal circumstances are low voltages, +3.3V, +5v, +12V, -5V & -12V. The power supply will have to be plugged in. Set your meter for the voltage scale and place the black test prod into the negative on one of the output connectors(usually the black lead) and put the red test prod into the red or yellow. You should get a reading of +3.3V, +5v, +12V, -5V or -12V. Without a power supply schematic you won’t know which pins should be which voltage, but if you get no voltage or some very high voltage, then you have a problem. Do this for all the connectors coming out of the power supply. Power supplies are inexpensive. I would say if you suspect a power supply is bad, just replace it. Even with 30 years experience, I don’t even bother repairing them.
Testing a motherboard can actually damage it if it wasn’t damaged to begin with. If you start poking a mulitmeter probe around a motherboard you could do more harm then good if you do not know what you are doing. Just use your nose and eyes for this one. A good source for multimeters is
www.oselectronics.com. They carry a wide selection of meters and tools for all kinds of electronic repairs. Good luck.