If you want to measure ESR, you'll have to buy an ESR meter ... usually a separate item, the function not available on most inexpensive testers.
If you want to accurately measure capacitor values, you'll need a digital cap meter such as the B+K 830. The "C" range of a digital multimeter will work, but won't be as accurate or have the measurement range of a decent dedicated digital cap meter.
If you want to test a cap for voltage breakdown, you'll need an older capacitance bridge such as an EICO 950B or Knight-Kit KG-670 or any one of three or four Heathkit models. One of these will also measure cap values, but their accuracy is usually limited to about ±5%.
If you want to measure leakage on high-voltage capacitors, you're almost going to have to use a megger or a semiconductor curve tracer such as a Tektronix 576 that's capable of putting out 1000 volts and measuring in the nanoamp region.