The appeal of quadruped robots lies not only in their ability to mimic the diverse and agile locomotion of animals but also in their potential to integrate with human artistic expression to achieve complex multi-skill movements. Continuous multi-skill motion in quadruped robots requires the realization of diverse, continuous, and long-horizon behaviors, involving a broader state space and more complex motion generation and transitions. This presents significant challenges, including sparse rewards, incomplete data, long-horizon motion planning, and the design of fine-grained motion transitions. In this work, we categorize quadruped robot skills into three types: rhythmic motions, expressive motions, and high-dynamic motions and generate reference trajectories for each category using central pattern generators, animation design, and motion capture, respectively. We then design an asymmetric neural network architecture and employ an imitation–reinforcement learning algorithm to train policies for generating these three types of motions. By composing multiple motion skill trajectories, we avoid long-horizon motion planning; by leveraging reinforcement learning, we enable smooth and continuous skill transitions; and by introducing a two-stage reference state initialization curriculum, the robot is able to switch from arbitrary states to the target motion skill. Moreover, during training, the policy imitates only key characteristics of the reference motions rather than strictly tracking fixed trajectories, making it more robust. Finally, we achieve robust motion skill generation and seamless transitions on a quadruped robot equipped with a 6-DoF manipulator, validating the effectiveness and feasibility of the proposed multi-skill generation and transition method.