Background: Goat milk has gained global attention for its unique nutritional properties and potential health benefits. Advancements in functional genomic technologies have significantly progressed genetic research on milk composition traits in dairy goats. Results: This review summarizes various research methodologies applied in this field. Genome-wide association studies (GWAS) have identified genomic regions associated with major milk components, with the diacylglycerol acyltransferase 1 (DGAT1) gene and casein gene cluster consistently linked to milk composition traits. Transcriptomics has revealed gene expression patterns in mammary tissue across lactation stages, while the role of non-coding RNAs (such as miRNAs and circRNAs) in regulating milk composition has been confirmed. Proteomic and metabolomic studies have not only helped us gain a more comprehensive understanding of goat milk composition characteristics but have also provided crucial support for the functional validation of genes related to milk components. The integration of multi-omics data has emerged as an effective strategy for elucidating complex regulatory networks from a systems biology perspective. Conclusions: Despite progress, challenges remain, including refining reference genomes, collecting large-scale phenotypic data, and conducting functional validations. Future research should focus on improving reference genomes, expanding study populations, investigating functional milk components, exploring epigenetic regulation and non-coding RNAs, and studying microbiome–host genome interactions. These efforts will inform more precise genomic and marker-assisted selection strategies, advancing genetic improvements in milk composition traits in dairy goats.