Bronchoscopy with transbronchial biopsy is a minimally invasive and effective method for early lung cancer intervention. Robot-assisted bronchoscopy offers improved precision, spatial flexibility, and reduced risk of cross-infection. This paper introduces a novel teleoperated robotic bronchoscopy system and a three-stage procedure designed for robot-assisted bronchoscopy. The robotic mechanism enables a clinical practice similar to traditional bronchoscopy, augmented by the control of a novel variable stiffness catheter for tissue sampling. A rapid prototype of the robotic system has been fully developed and validated through in-vivo experiments. The results demonstrate the potential of the proposed robotic bronchoscopy system and variable stiffness catheter in enhancing accuracy and safety during bronchoscopy procedures.