We employ a characterization of linguistic complexity from psycholinguistic and language acquisition research to develop data-driven curricula to understand the underlying linguistic knowledge that models learn to address NLP tasks. The novelty of our approach is in the development of linguistic curricula derived from data, existing knowledge about linguistic complexity, and model behavior during training. By analyzing several benchmark NLP datasets, our curriculum learning approaches identify sets of linguistic metrics (indices) that inform the challenges and reasoning required to address each task. Our work will inform future research in all NLP areas, allowing linguistic complexity to be considered early in the research and development process. In addition, our work prompts an examination of gold standards and fair evaluation in NLP.