The optimization of machining processes is a deciding factor when increasing productivity and ensuring product quality. The response characteristics, such as surface roughness, material removal rate, tool wear, and cutting time, of the finish turning process have been simultaneously optimized. We used the Taguchi-based design of experiments L9(34) in this study to test and find the best values for process parameters like cutting speed, feed rate, depth of cut, and nose radius. The Taguchi-based multi-objective grey relational approach (GRA) method was used to address the turning problem of Inconel 718 alloy to increase productivity, i.e., by simultaneously minimizing surface roughness, tool wear, and machining time. GRA and the S/N ratio derived from the Taguchi approach were utilized to combine many response characteristics into a single response. The grey relational grade (GRG) produces results such as estimations of the optimal level of input parameters and their proportional significance to specific quality characteristics. By employing ANOVA, the significance of parameters with respect to individual responsibility and the overall quality characteristics of the cutting process were ascertained. The single-objective optimization yielded the following results: minimal surface roughness of 0.167 µm, tool wear of 44.65 µm, minimum cutting time of 19.72 s, and maximum material speed of 4550 mm3/min. While simultaneously optimizing the Inconel 718 superalloy at a cutting speed of 100 m/min, depth of cut of 0.4 mm, feed rate of 0.051 mm/rev, and tool nose radius of 0.4 mm, the results of the multi-objective optimization showed that all investigated response characteristics reached their optimal values (minimum/maximum). To validate the results, confirmatory experiments with the most favorable outcomes were conducted and yielded a high degree of concurrence.