Research in child-robot interactions suggests that engaging in "care-taking" of a social robot, such as tucking the robot in at night, can strengthen relationships formed between children and robots. In this work, we aim to better understand and explore the design space of caretaking activities with 10 children, aged 8--12 from eight families, involving an exploratory design session followed by a preliminary feasibility testing of robot caretaking activities. The design sessions provided insight into children's current caretaking tasks, how they would take care of a social robot, and how these new caretaking activities could be integrated into their daily routines. The feasibility study tested two different types of robot caretaking tasks, which we call connection and utility, and measured their short term effects on children's perceptions of and closeness to the social robot. We discuss the themes and present interaction design guidelines of robot caretaking activities for children.