Automating dexterous, contact-rich manipulation tasks using rigid robots is a significant challenge in robotics. Rigid robots, defined by their actuation through position commands, face issues of excessive contact forces due to their inability to adapt to contact with the environment, potentially causing damage. While compliance control schemes have been introduced to mitigate these issues by controlling forces via external sensors, they are hampered by the need for fine-tuning task-specific controller parameters. Learning from Demonstrations (LfD) offers an intuitive alternative, allowing robots to learn manipulations through observed actions. In this work, we introduce a novel system to enhance the teaching of dexterous, contact-rich manipulations to rigid robots. Our system is twofold: firstly, it incorporates a teleoperation interface utilizing Virtual Reality (VR) controllers, designed to provide an intuitive and cost-effective method for task demonstration with haptic feedback. Secondly, we present Comp-ACT (Compliance Control via Action Chunking with Transformers), a method that leverages the demonstrations to learn variable compliance control from a few demonstrations. Our methods have been validated across various complex contact-rich manipulation tasks using single-arm and bimanual robot setups in simulated and real-world environments, demonstrating the effectiveness of our system in teaching robots dexterous manipulations with enhanced adaptability and safety. Code available at: https://github.com/omron-sinicx/CompACT