There isn't really any easy way to test a power supply without an oscilloscope. Some power supplies may be weak and supply insufficient voltage, or perhaps a bit much volage, which may provide for unreliable operation, but it is the occasional voltage spike which will destroy motherboards and other expensive equipment. These spikes are usually too brief to be noticed on any standard voltmeter, thus the need for an oscilloscope, which is probably way out of your price range.
The best way to avoid trouble is to never use any generic power supply, and only use reliable brands such as enermax (my favorite) and antec. You can use a generic (and therefore suspect) ATX power supply as a bench supply (put a .5 A or so load on the 5v rail for reliable operation) or in equipment that doesn't necesarily need clean power. Or you could sell your generic ATX supplies to a person that doesn't have high standards on the computer equipment they use. Check around computing sites for other good brands if you like, but I personally will only purchase power supplies of the two mentioned brands.
Kizap is right about how to turn it on, take a paperclip or other convenient conducting object, insert it into pin 14 and 15 on the main plug, or pin 14 and any pin that has a black wire running to it. The power supplies own fan should start running now. Put a load on the supply, either some high wattage resistors or some spare fans, and measure the voltages to get an idea how accurate the supply is.