Training a bipedal character to play basketball and interact with objects, or a quadruped character to move in various locomotion modes, are difficult tasks due to the fast and complex contacts happening during the motion. In this paper, we propose a novel framework to learn fast and dynamic character interactions that involve multiple contacts between the body and an object, another character and the environment, from a rich, unstructured motion capture database. We use one-on-one basketball play and character interactions with the environment as examples. To achieve this task, we propose a novel feature called local motion phase, that can help neural networks to learn asynchronous movements of each bone and its interaction with external objects such as a ball or an environment. We also propose a novel generative scheme to reproduce a wide variation of movements from abstract control signals given by a gamepad, which can be useful for changing the style of the motion under the same context. Our scheme is useful for animating contact-rich, complex interactions for real-time applications such as computer games.