Uncertainty in control and perception poses challenges for autonomous vehicle navigation in unstructured environments, leading to navigation failures and potential vehicle damage. This paper introduces a framework that minimizes control and perception uncertainty to ensure safe and reliable navigation. The framework consists of two uncertainty-aware models: a learning-based vehicle dynamics model and a self-supervised traversability estimation model. We train a vehicle dynamics model that can quantify the epistemic uncertainty of the model to perform active exploration, resulting in the efficient collection of training data and effective avoidance of uncertain state-action spaces. In addition, we employ meta-learning to train a traversability cost prediction network. The model can be trained with driving data from a variety of types of terrain, and it can online-adapt based on interaction experiences to reduce the aleatoric uncertainty. Integrating the dynamics model and traversability cost prediction model with a sampling-based model predictive controller allows for optimizing trajectories that avoid uncertain terrains and state-action spaces. Experimental results demonstrate that the proposed method reduces uncertainty in prediction and improves stability in autonomous vehicle navigation in unstructured environments.