A detailed understanding of the interaction between a drug candidate molecule and its target is essential for the development, optimization, and efficacy prediction of a drug. Kinetic parameters such as the association rate and residence time of a molecule have been shown to better correlate with in vivo efficacy than more commonly used thermodynamic parameters. Efficient and accurate computational predictions of these quantities are therefore of great interest for their potential to inform and improve the development of novel pharmaceuticals. In this dissertation, I present the development and application of a multiscale molecular simulation approach which combines molecular dynamics and Brownian dynamics simulations with the theory of milestoning to efficiently calculate protein-ligand binding and unbinding rates. I begin with an overview of many of the existing multiscale simulation approaches for studying drug-protein binding. Then I present the methodology we have developed, Simulation Enabled Estimation of Kinetic Rates (SEEKR), and demonstrate its effectiveness for predicting the association and dissociation rates of the inhibitor, benzamidine, to the trypsin protein; a common model system. I then present the effectiveness of our multiscale milestoning approach for rank-ordering a series of chemically diverse ligands to the model system β-cyclodextrin. This study includes a direct comparison of both efficiency and accuracy to long timescale molecular dynamics simulations and also outlines best practices for the use of our approach and the assessment of sampling convergence. Finally, I present the implementation of a new milestoning algorithm, Markovian Milestoning with Voronoi Tesselations, in our multiscale methodology to significantly decrease the simulation cost of kinetics calculations, improve the assessment of sampling convergence, and provide a framework for the future development of additional capabilities with the SEEKR method. This study also includes the development and deployment of our toolkit along with documentation and tutorials to facilitate its use and continued improvement by the scientific community.