There are many methods of testing capacitor. It depends on individual as what type of testing equipment they want to use. Some prefer the old way while some are using the latest and expensive equipment.
You can use either one of these meters to test capacitors ie; the analog meter, digital capacitance meter, ESR meter or even an insulation tester. All of these meters have it own strenght of checking a capacitor. If you have some budget and would like to have a meter that perform the complete four tests, then i guess SENCORE LC103 capacitor and inductor analyzer is most suitable to you. Besides testing all kinds of capacitors, it also have the feature to test coils/inductors too, like ringing a flyback transformer and etc. However the price is a bit high and in the range of US2500.00.