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The ability to engineer the kinetic and thermodynamic processes by which nanoparticles and colloids form crystals would open new possibilities for materials design. Simulations, coupled with data science and machine learning, can open new frontiers towards obtaining this kind of control over matter at the nanoscale. We need novel approaches in theory and computation to bridge the perspectives of forward design (predicting the properties of a material from its components and their interactions) and inverse design (predicting components and interactions that produce a desired set of material properties). In Chapter I, I provide background about materials design via self-assembly and outline the questions I address in this dissertation. In Chapter II, I present mathematical and physical motivation for a new structural descriptor, the Continuous Topological Order Parameter (CTOP), for analyzing crystallization pathways during self-assembly from a microscopic (particle-local) perspective. The CTOP is comprised of Minkowski Structure Metrics and captures continuous deformations in particles' local environments, which we express as a high-dimensional feature space. This space is coupled with the topology-preserving Uniform Manifold Approximation and Projection (UMAP) dimensionality reduction algorithm to produce a continuous mapping from particles' local environments into interpretable self-assembly pathways. In Chapter III, I apply the CTOP method to a wide range of nanoparticle systems undergoing self-assembly. I directly investigate the CTOP manifold to illuminate aspects of the crystallization process in many types of simple and complex crystals. Hard Particle Monte Carlo and molecular dynamics simulations of pair potentials are analyzed, resulting in a diverse array of self-assembled crystal structures with unit cells ranging from one to 54 particles. I apply unsupervised learning methods to the pathways revealed by the CTOP method, enabling identification of particles' local environments in self-assembling systems including Frank-Kasper phases and decagonal quasicrystals. Next, I show comparisons of pathways with this method, including solid-solid phase transitions. The chapter closes with a discussion of this method's implications for the study of self-assembly, and an outlook on how continuous feature spaces may inspire future analyses and engineering applications. Chapter IV presents a study of machine learning applied to photonic crystals developed in an effort to accelerate photonic materials design. I discuss the development of convolutional neural networks and equivariant neural networks for predicting photonic properties. I conclude with a discussion of photonic densities of states and equivariance in physics-based machine learning, which may provide further insight on this challenging problem. Chapter V covers my contributions as a core developer and maintainer of the freud library, an open-source software package used for data analysis in this dissertation. I describe the use of the freud library in machine learning pipelines and data visualization, and summarize publications using freud (33 to date) across the fields of soft matter, statistical mechanics, and particle-based simulation. In Chapter VI, I discuss my contributions as a core developer and maintainer of the open-source signac data management framework, which helps researchers execute reproducible computational studies, scaling from laptops to supercomputers and emphasizing portability and fast prototyping. I describe the framework's data model, HDF5 data stores for large numerical arrays, enhancements to performance and scalability, and the signac-dashboard application for data visualization. Finally, I conclude with a summary of the work presented in this dissertation and insights for future research in the field of self-assembly pathways, photonics, and software designed for computational researchers.