Scientific missions to Lagrangian points have the potential to enhance the understanding of the universe and to accelerate the exploration of space. A typical mission design to an orbit around the Lagrangian points from the Earth involves two steps. In the first step, an orbit with prescribed geometrical characteristics is designed and in the second step, an optimal transfer trajectory to the orbit from an Earth parking orbit is designed. In the conventional approach to generate the preliminary design, the model of Circular Restricted Three Body Problem (CRTBP) framework is adopted utilizing differential correction (DC) and the transfer trajectory design using manifold theory. In this research, the preliminary design of orbits and transfer to them from the Earth are proposed to be executed under the framework of Elliptic Restricted Three Body Problem (ERTBP) utilizing an evolutionary optimization technique called Differential Evolution (DE). Various periodic orbits and quasi-periodic orbits in the Sun-Earth-spacecraft and Earth-Moon-spacecraft restricted three body systems and transfer trajectories to them from the Earth are generated without leveraging the manifold theory. For the transfer trajectory design in the Earth-Moon system, the proposed two-impulse methodology avoids the bridge segment and generates optimal trajectories with significantly lower flight durations compared to the manifold theory. The preliminary numerical designs are extended to high fidelity ephemeris models. In the ephemeris model, the generated quasi-halo orbits in the Sun-Earth system do not need any theoretical design maneuvers for about five years. For the mission design in the Sun-Earth system, it is substantively concluded that preliminary design using the ERTBP framework does not provide significant advantages over the CRTBP framework due to the near-circular nature of the Earth’s orbit around the Sun. The differential evolution technique is found to be very versatile in solving Lagrangian point mission design problems and avoids many complexities associated with the differential correction based technique. However, the DE based schemes are found to be computationally more intensive. The outcomes of this research can provide valuable methodologies and insights that can significantly enhance the effectiveness of future Lagrangian point missions, thus paving the way for further exploration and scientific discoveries in space.