There has been growing consumer interest in sheep and goat milk products as alternatives to cow milk products. The physicochemical characteristics of milk vary not only between ruminant species, but also during different seasons; they determine the nutritional quality and processing properties of the milk. In this study, we characterized sheep and goat milks from New Zealand over the seasons for their composition (macronutrients, macro- and micro-minerals, fatty acids, and proteins) and physicochemical properties (e.g., ionic calcium, fat globule size, casein micelle size, viscosity, and melting behavior of milk fat). Heat-induced (95 °C for 5 min) protein interactions and changes in the physical properties of the milks were also investigated. The compositional and structural features of sheep and goat milks were identified and compared with those reported for cow milk. Seasonal variations in the milk characteristics were more pronounced for sheep milk than goat milk and were probably affected by the production systems. Sheep milk, particularly in the late season, had the largest heat-induced increases in casein micelle size and viscosity, probably arising from the greater casein-whey protein and casein-casein interactions during heat treatment. This study provides comprehensive information on the properties of sheep and goat milks and highlights the interaction effects between species, season, and processing.