Bird-sized flapping-wing robots offer significant potential for agile flight in complex environments, but achieving agile and robust trajectory tracking remains a challenge due to the complex aerodynamics and highly nonlinear dynamics inherent in flapping-wing flight. In this work, a learning-based control approach is introduced to unlock the versatility and adaptiveness of flapping-wing flight. We propose a model-free reinforcement learning (RL)-based framework for a high degree-of-freedom (DoF) bird-inspired flapping-wing robot that allows for multimodal flight and agile trajectory tracking. Stability analysis was performed on the closed-loop system comprising of the flapping-wing system and the RL policy. Additionally, simulation results demonstrate that the RL-based controller can successfully learn complex wing trajectory patterns, achieve stable flight, switch between flight modes spontaneously, and track different trajectories under various aerodynamic conditions.