Colostrum is a major source of immunity in ruminants. It allows the transfer of antibodies from the mother to the fetus, and it is the exclusive source of nutrients for the newborn. The objectives of this study were (i) to characterize the macro and the micro composition of colostrum; (ii) to analyze the antioxidant capacity, fatty acid profile, and essential and toxic mineral content of colostrum; and (iii) to compare FA profiles and the amount of trace elements between colostrum and mature milk. For these purposes, samples of colostrum and milk were collected from a representative sample of animals from eight sheep dairy farms in the north of Sardinia (Italy). Fat, proteins, and seven essential and toxic minerals were measured in all samples of colostrum and milk. Furthermore, the FA profile was also measured in both matrices, while total antioxidant capacity was measured only in colostrum samples. The average amounts of fat and protein (TP) concentration in colostrum were 7.8% and 16%, respectively. Additionally, an average amount of 40 ± 20 g dm−3 was found for immunoglobulin G (IgG). As regards the antioxidant capacity of colostrum, a large variation was observed between samples from different farms for test 2, 2′-azino-bis(3-ethylbenzthiazoline-6-sulfonic acid) (ABTS), which was 30 ± 10% (mean ± standard deviation). High levels of selenium (Se), zinc (Zn), and copper (Cu) were found in colostrum (200 µg kg−1, 25,000 µg kg−1, and 1200 µg kg−1, respectively). A strong positive correlation between TP and IgG was observed (r = 91%). In colostrum, the amount of IgG is positively correlated with Se and Zn, as they are essential minerals to the immune system. The FA profile demonstrated higher levels of medium and long chain fatty acids in colostrum than in mature milk, and this is mainly true for arachidonic acid (ARA), ecosapentaenoic acid (EPA), docosapentaenoic acid (DPA), and docosahexaenoic acid (DHA). This study provided new information on the quality of colostrum in Sarda dairy sheep and showed the different composition of fatty acids between colostrum and mature milk.