This study introduces an innovative strategy for addressing the Home Healthcare and Dial-a-Ride Problem (HHCDAP) concerning the transportation of medical staff and patients, taking into account the stochastic nature of service and travel times. The problem involves assigning suitable medical staff to patients and clients, determining the order of visits, and identifying opportunities for medical staff and patients to share trips. We propose two objective functions to minimize travel time for drivers and medical staff. This problem adheres to numerous constraints, including maximum work duration, maximum waiting time, professional qualifications, and vehicle capacity limitations. We test our approach on a small-scale instance to understand the trade-offs between minimizing drivers' travel time and minimizing the travel and waiting times of medical staff and patients. Our results indicate that the proposed strategy enhances the efficiency of transporting medical staff and patients.