<p>Biological systems such as mountain goats and felines exhibit remarkable agility and adaptability when traversing complex terrains. Inspired by these capabilities, quadruped robots have been developed to mimic legged locomotion and improve mobility over uneven environments. To further enhance locomotion efficiency and terrain versatility, wheeled-legged robots integrate wheels and legs into a hybrid platform, enabling both high-speed traversal and robust ground contact in unstructured terrain. However, planning coordinated locomotion across diverse terrains remains challenging due to the nonlinear dynamics, complex terrain contact constraints, and multimodal locomotion capabilities. In this paper, we propose a real-time, integrated planning framework that jointly optimizes gait scheduling, footstep placement, and whole-body motion trajectories. Our method adopts a two-stage approach. First, a sampling-based planner generates candidate gait sequences and nominal footstep targets based on terrain features and kinematic feasibility. Second, a constrained trajectory optimizer reformulates the planning problem as a Quadratic Programming (QP) task to compute dynamically feasible base trajectories and corresponding ground reaction forces. This hybrid formulation balances planning efficiency and physical realism. The planned trajectories and contact forces are tracked using a hierarchical control architecture combining Model Predictive Control (MPC) and Whole-Body Control (WBC), enabling fast and stable execution on real hardware. Simulation and real-world experiments demonstrate that our approach enables adaptive gait transitions and improves terrain adaptability compared to traditional planners.</p>