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This paper investigates propulsion options and trajectory designs for a manned mission from Earth to Mars, followed by a mission to Ceres. It evaluates various propulsion technologies, including traditional chemical propulsion, nuclear thermal propulsion (NTP), and electric propulsion systems such as ion thrusters and Hall effect thrusters. The analysis highlights the limitations of chemical propulsion in terms of energy efficiency and payload capacity while underscoring the potential of NTP to reduce travel time and improve crew safety due to its higher specific impulse. The study also examines the feasibility of solar and nuclear electric propulsion, focusing on their high efficiency and suitability for extended missions, and considers innovative concepts like solar sails and fusion propulsion for their theoretical advantages in thrust and velocity. Low-thrust trajectories are analyzed for their ability to minimize overall delta-V requirements, which is essential for mission success. For the Mars mission, chemical propulsion systems currently under development are evaluated alongside electric propulsion powered by nuclear reactors, which could significantly reduce travel times and propellant needs. The additional challenges of propelling a spacecraft to Ceres, a more distant destination in the asteroid belt, shift the focus to electric propulsion options, particularly advanced nuclear-electric systems, which offer the potential to enable human exploration within a reasonable timeframe. The paper concludes that the optimal propulsion system for a manned mission to Mars and Ceres requires balancing travel time, technical feasibility, and crew safety, emphasizing the necessity for further development of advanced electric propulsion technologies to facilitate ambitious human deep-space exploration.