The enhanced mobility brought by legged locomotion empowers quadrupedal robots to navigate through complex and unstructured environments. However, optimizing agile locomotion while accounting for the varying energy costs of traversing different terrains remains an open challenge. Most previous work focuses on planning trajectories with traversability cost estimation based on human-labeled environmental features. However, this human-centric approach is insufficient because it does not account for the varying capabilities of the robot locomotion controllers over challenging terrains. To address this, we develop a novel traversability estimator in a robot-centric manner, based on the value function of the robot's locomotion controller. This estimator is integrated into a new learning-based RGBD navigation framework. The framework employs multiple training stages to develop a planner that guides the robot in avoiding obstacles and hard-to-traverse terrains while reaching its goals. The training of the navigation planner is directly performed in the real world using a sample efficient reinforcement learning method that utilizes both online data and offline datasets. Through extensive benchmarking, we demonstrate that the proposed framework achieves the best performance in accurate traversability cost estimation and efficient learning from multi-modal data (including the robot's color and depth vision, as well as proprioceptive feedback) for real-world training. Using the proposed method, a quadrupedal robot learns to perform traversability-aware navigation through trial and error in various real-world environments with challenging terrains that are difficult to classify using depth vision alone. Moreover, the robot demonstrates the ability to generalize the learned navigation skills to unseen scenarios. Video can be found at https://youtu.be/RSqnIWZ1qks.