Goat meat represents a valuable source of high-quality protein and healthy lipids, although its consumption remains limited in Europe. This study aimed to evaluate the qualitative–quantitative changes in the nutritional, rheological, and sensorial characteristics of meat (Quadriceps femoris and Longissimus thoracis et lumborum muscles) from three different autochthonous goat breeds (Garganica, Derivata di Siria, and Capra di Potenza) and a cosmopolitan, genetically selected one (Saanen), reared in Basilicata (Italy), during a 7-day wet aging process. Forty kids (10 per breed) were slaughtered at 50 ± 3 days, and meat samples were vacuum-aged at 4 ± 1 °C and analyzed at 0, 3, and 7 days. Data showed that genotype was the main factor influencing meat quality, while wet aging mainly improved rheological parameters, particularly in LTL muscles. Notably, Capra di Potenza exhibited the most favorable fatty acid profile, with lower atherogenic (average values of 0.80 in LTL and 0.92 in QF) and thrombogenic (average values of 1.49 in LTL and 1.59 in QF) indices, whereas Derivata di Siria showed the greatest oxidative stability (average values of 0.060 in LTL and 0.036 in QF). Overall, local breeds of kids’ groups produced more tender and aromatic meat than Saanen. These findings highlight the potential of native goat breeds for premium meat production and suggest an effective post-mortem aging technique to enhance their quality, promoting the diffusion of niche products as well as biodiversity preservation.