We present curiosity-driven, autonomous acquisition of tactile exploratory skills on a biomimetic robot finger equipped with an array of microelectromechanical touch sensors. Instead of building tailored algorithms for solving a specific tactile task, we employ a more general curiosity-driven reinforcement learning approach that autonomously learns a set of motor skills in absence of an explicit teacher signal. In this approach, the acquisition of skills is driven by the information content of the sensory input signals relative to a learner that aims at representing sensory inputs using fewer and fewer computational resources. We show that, from initially random exploration of its environment, the robotic system autonomously develops a small set of basic motor skills that lead to different kinds of tactile input. Next, the system learns how to exploit the learned motor skills to solve supervised texture classification tasks. Our approach demonstrates the feasibility of autonomous acquisition of tactile skills on physical robotic platforms through curiosity-driven reinforcement learning, overcomes typical difficulties of engineered solutions for active tactile exploration and underactuated control, and provides a basis for studying developmental learning through intrinsic motivation in robots.