How does the three-part Catholic sacrament of baptism relate to the Catholic belief in the Trinity?
If you mean by “three-part Catholic sacrament of baptism” the three sacraments of initiation (baptism, confirmation, eucharist)… the “threeness” of the Trinity and the “threeness” of the initiation process are not related. However, one being initiated into the Catholic faith IS being initiated into a Trinitarian faith.