The cellular cytoskeleton is essential in proper cell function as well as in organism development. These filaments represent the roads along which most protein transport occurs inside cells. I will discuss several examples where questions about filament-motor protein interactions require the development of novel mathematical modeling, analysis, and simulation. In the development of egg cells into embryos, RNA molecules bind to and unbind from cellular roads called microtubules, switching between bidirectional transport, diffusion, and stationary states. Since models of intracellular transport can be analytically intractable, asymptotic methods are useful in understanding effective cargo transport properties as well as their dependence on model parameters. We consider these models in the framework of partial differential equations as well as stochastic processes and derive large time properties of cargo movement for a general class of problems. The proposed methods have applications to macroscopic models of protein localization and microscopic models of cargo movement by teams of motor proteins. I will also discuss an agent-based modeling and data analysis framework for understanding how actin filaments and myosin motors interact to form contractile ring channels essential in development. In particular, we propose tools drawing from topological data analysis to analyze time-series data of filament network interactions and illustrate the impact of key parameters on significant ring emergence, thus giving insight into formation and maintenance of these biological channels.