Despite recent advancements in torque-controlled tactile robots, integrating them into manufacturing settings remains challenging, particularly in complex environments. Simplifying robotic skill programming for non-experts is crucial for increasing robot deployment in manufacturing. This work proposes an innovative approach, Vision-Augmented Unified Force-Impedance Control (VA-UFIC), aimed at intuitive visuo-tactile exploration of unknown 3D curvatures. VA-UFIC stands out by seamlessly integrating vision and tactile data, enabling the exploration of diverse contact shapes in three dimensions, including point contacts, flat contacts with concave and convex curvatures, and scenarios involving contact loss. A pivotal component of our method is a robust online contact alignment monitoring system that considers tactile error, local surface curvature, and orientation, facilitating adaptive adjustments of robot stiffness and force regulation during exploration. We introduce virtual energy tanks within the control framework to ensure safety and stability, effectively addressing inherent safety concerns in visuo-tactile exploration. Evaluation using a Franka Emika research robot demonstrates the efficacy of VA-UFIC in exploring unknown 3D curvatures while adhering to arbitrarily defined force-motion policies. By seamlessly integrating vision and tactile sensing, VA-UFIC offers a promising avenue for intuitive exploration of complex environments, with potential applications spanning manufacturing, inspection, and beyond.