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In recent years, Taiwan has been facing water shortages due to the impact of climate change, which has resulted in many serious drought events, especially in southern Taiwan. Long-term records from 25 rainfall stations and 17 groundwater stations in the southern Taiwan basin were used in this study. We used the Standardized Precipitation Index (SPI) and the Standardized Groundwater Level Index (SGI) and employed the first-order Markov chain model and wavelet transform to determine the drought characteristics and propagation, including the steady-state probabilities of drought events and the mean duration for each station. The Drought Index (DI) was also used to investigate the effects of rainfall on groundwater drought. The results show that the steady-state probability of the meteorological drought in the Yanshui River basin in southern Taiwan is higher than that in other basins. The area with the longer mean duration is located in the Yanshui River basin and the Erren River basin, and overall, the mean duration ranges from 3 to 7 months. In addition, the results from the drought proneness analysis indicated that when rainfall causes a longer drought duration, there will be a higher degree of proneness to groundwater drought in the future. Finally, the results show that the mean duration of groundwater droughts are longer than those of meteorological droughts. The results of the wavelet analysis revealed a positive correlation at long-term scales, which may be related to large-scale atmospheric circulation. The information from this research could be used as a reference for water resource management in the future.