One of the most widely used additive manufacturing (AM) methods is Fused Filament Fabrication (FFF), which can produce complex geometry parts. In this process, a continuous filament of thermoplastic material is deposited layer by layer to make the final piece. One of the essential goals in the production of parts with this method is to produce parts with high mechanical properties and excellent geometrical accuracy at the same time. Among the various methods used to improve the desired properties of produced parts is to determine the optimum process parameters in this process. This paper investigates the effect of different process parameters on four essential parameters: chamber temperature, Printing temperature, layer thickness, and print speed on cylindricity, circularity, strength, Young’s modulus, and deformation by Gray Relational Analysis method simultaneously. Taguchi method was used to design the experiments, and the PA6 cylindrical parts were fabricated using a German RepRap X500® 3D printer. Then the GRG values were calculated for all experiments. In the 8th trial, the highest value of GRG was observed. Then, to discover the optimal parameters, the GRG data were analyzed using ANOVA and S/N analysis, and it was determined that the best conditions for enhancing GRG are 60 °C in the chamber temperature, 270 °C in the printing temperature, 0.1 mm layer thickness, and 600 mm/min print speed. Finally, by using optimal parameters, a verification test was performed, and new components were investigated. Finally, by comparing the initial GRG with the GRG of the experiment, it was discovered that the GRG value had improved by 14%.