Unmanned aerial vehicles represent a new frontier in a wide range of monitoring and research applications. To fully leverage their potential, a key challenge is planning missions for efficient data acquisition in complex environments. To address this issue, this article introduces a general informative path planning framework for monitoring scenarios using an aerial robot, focusing on problems in which the value of sensor information is unevenly distributed in a target area and unknown a priori. The approach is capable of learning and focusing on regions of interest via adaptation to map either discrete or continuous variables on the terrain using variable-resolution data received from probabilistic sensors. During a mission, the terrain maps built online are used to plan information-rich trajectories in continuous 3-D space by optimizing initial solutions obtained by a coarse grid search. Extensive simulations show that our approach is more efficient than existing methods. We also demonstrate its real-time application on a photorealistic mapping scenario using a publicly available dataset and a proof of concept for an agricultural monitoring task.