Probabilistic forecasting and nowcasting of infectious disease targets have been an important tool used by public health officials, researchers, and stakeholders, for studying disease dynamics, monitoring outbreaks, and informing public health response to epidemics. Improving the quality of infectious disease forecasts and our understanding of their behavior and accuracy is a challenging but crucial task. This dissertation contributes toward that objective by studying multiple facets of infectious disease forecasting and nowcasting, including methods for generating forecasts and nowcasts and how their outputs can be analyzed and evaluated. Firstly, we present the adaptation and application of existing calibration methods for ensemble forecasts and the comparison of their performance in an application of seasonable influenza forecasting. Secondly, we propose an approximation of the Cramér distance as a method to quantify similarity of probabilistic forecasts and illustrate the utility of this method through analyses of the characteristics of forecasts of COVID-19 mortality. Lastly, we explore a machine learning approach and a relative growth advantage model to generate nowcasts of SARS-CoV-2 clade proportions in the U.S. and evaluate their performance.