Self-driving vehicles must perceive and predict the future positions of nearby actors in order to avoid collisions and drive safely. A learned deep learning module is often responsible for this task, requiring large-scale, high-quality training datasets. As data collection is often significantly cheaper than labeling in this domain, the decision of which subset of examples to label can have a profound impact on model performance. Active learning techniques, which leverage the state of the current model to iteratively select examples for labeling, offer a promising solution to this problem. However, despite the appeal of this approach, there has been little scientific analysis of active learning approaches for the perception and prediction (P&P) problem. In this work, we study active learning techniques for P&P and find that the traditional active learning formulation is ill-suited for the P&P setting. We thus introduce generalizations that ensure that our approach is both cost-aware and allows for fine-grained selection of examples through partially labeled scenes. Our experiments on a real-world, large-scale self-driving dataset suggest that fine-grained selection can improve the performance across perception, prediction, and downstream planning tasks.