Green machining strategies in the manufacturing sector help to maintain the product value by considering the environmental impacts. Also, improvisation in the quality contribution of the parts can minimize the environmental consequences by improving resource efficiency, specifically in terms of coolants used in machining. Certain hazardous impacts have been witnessed because of longer exposure to such a machining environment. To address it, many researchers have concentrated on providing a healthy machining environment either by introducing dry machining or by minimum quantity lubrication (MQL). The proposed study addresses this context. The influence of these tactics on the attained surface quality of Al-6063 is quantified in this paper in terms of surface integrity (Ra) and removal rate of material (MRR). The study involves single-response optimization using the Taguchi design and multiresponse optimization using grey relational analysis (GRA). The results reveal that the depth of cut (Dc) and spindle speed (Ss) have the greatest impact on Ra and MRR. The machinability of Al-6063 is examined by considering the key machinability parameters, such as the spindle speed (Ss), feed rate (Fr), and the depth of cut (Dc), to arrive at the best possible surface roughness and removal rate of the material. As a typical Taguchi approach cannot perform multiresponse optimization, grey relational analysis is used. The grey relational analysis combined with Taguchi gives a novel methodology for multioptimization. The entire study is performed in dry condition and under minimum quantity lubrication. The results suggest that the responses are highly influenced by the depth of cut and spindle speed.