Inside cells, axonal and dendritic transport by motor proteins is a process that is responsible for supplying cargo, such as vesicles and organelles, to support neuronal function. Motor proteins achieve transport through a cycle of chemical and mechanical processes. Particle tracking experiments are used to study this intracellular cargo transport by recording multi-dimensional, discrete cargo position trajectories over time. However, due to experimental limitations, much of the mechanochemical process cannot be directly observed, making mathematical modeling and statistical inference an essential tool for identifying the underlying mechanisms. The cargo movement during transport is modeled using a switching stochastic differential equation framework that involves classification into one of three proposed hidden regimes. Each regime is characterized by different levels of velocity and stochasticity. The equations are presented as a state-space model with Markovian properties. Through a stochastic expectation-maximization algorithm, statistical inference can be made based on the observed trajectory. Regime predictions and particle location predictions are calculated through an auxiliary particle filter and particle smoother. Based on these predictions, parameters are estimated through maximum likelihood. Diagnostics are proposed that can assess model performance and therefore also be a form of model selection criteria. Model selection is used to find the most accurate regime models and the optimal number of regimes for a certain motor-cargo system. A method for incorporating a second positional dimension is also introduced. These methods are tested on both simulated data and different types of experimental data.