I recommend you review the Handbook of Drought Indicators and Indices, recently published by the World Meteorological Organization and the Global Water Partnership. In this handbook you will find the description of various indexes that you can calculate based on data of precipitation and temperature.
Another index that I do not see mentioned here is the Standardized Precipitation Index (SPI), introduced by McKee et al. 1993
The advantage of the SPI is you can explore droughts based on different time periods, such as past 3, 6, 12, 24, 48 months. I've used in this past to understand conditional relationships between drought and conflict, and the different lags reveal different possible mechanisms.