The intersection of dynamical systems and networks are used to model a huge variety of phenomena such as the spread of disease, multi-agent systems, opinions in social networks, and more. Many properties of these network phenomena can be understood by examining the eigenvalue spectrum of a matrix representation of the underlying graph. Using this intuition, this thesis explores the learning and control of network phenomena. First, I present techniques for matching individuals across correlated networks and learning the spectra of a graph matrix using only the sparse output measurements of a networked dynamical system with periodic inputs. Next, I present a data-driven framework for multi-task learning and non-linear control of epidemics. Finally, I propose a new architecture for signal processing on higher-order graphs, along with a new transferability bound on the performance of graph neural networks via spectral similarity. This transferability result is valid for arbitrary graphs regardless of their structure, resulting in the first bound on the transferability of a machine learning approach for higher-order graphs.