Concrete is the world’s most extensively manufactured material, with the integration of recycled materials becoming crucial for sustainable construction. This study investigates the mechanical properties of concrete mixes containing Recycled Aggregates (RA) and Portland Slag Cement (PSC) to understand their suitability for sustainable construction applications. Utilizing normal-strength concrete with a compressive strength of 35 MPa and a 0.45 water-to-cement ratio, fifteen mix designs were tested, varying RA and PSC replacement levels from 0 to 100% and 0% to 50%, respectively. Comprehensive mechanical testing, including compressive and tensile strength assessments at 7 and 28 days, alongside modulus of elasticity and stress–strain curve evaluations, was conducted. The results reveal that full replacement of Natural Aggregate (NA) with RA decreased compressive strength by 10% and modulus of elasticity by 20%. However, a partial replacement of 25% of Ordinary Portland Cement (OPC) with PSC, in the presence of RA, moderated the reduction of compressive strength to 0.92% and increased the modulus of elasticity by 3.7%. The optimum mix, consisting of 75% OPC, 25% PSC, and 50% RA, significantly enhances concrete’s mechanical properties, recommending it for reinforced concrete applications due to its potential environmental and structural benefits.