Data graphs are central to scientific communication, educational material, and news content, yet the building blocks of graph comprehension remain only partially understood. Our work provides an initial assessment of three questions: (1) how can the conceptual knowledge underlying graph understanding be characterized? (2) how do these abilities develop from childhood to adulthood? and (3) how are they related to perceptual abilities and higher-level graph understanding? We propose a hierarchical model of graphicacy organized around a set of conceptual components ranging from basic numerical and geometric precursors to the understanding of functional relationships. To test the model, 146 adults completed a novel conceptual graphicacy task, and a subset of 58 also completed a perceptual trend judgment task. Accuracy patterns across items and conceptual categories revealed systematic differences in component difficulty, broadly consistent with the proposed conceptual progression. Conceptual graphicacy skills, understood as the knowledge participants possess about graph conventions and relationships, were significantly predicted by perceptual sensitivity in detecting the trend in a scatterplot and with self-reported mathematical skills. To probe the development of conceptual graphicacy skills, 191 primary school children completed a shortened version of the task. Performance improved with age, revealing how the hierarchy of conceptual knowledge develops across school years. Conceptual graphicacy also significantly predicted performance on a higher-level graphical reasoning task in children. Together, these findings provide initial evidence linking perceptual, conceptual, and higher-level components of graphicacy within a common framework.