Costa learned the ru­di­ments of mu­sic from his ma­ter­nal grand­fa­ther, Gi­a­co­mo Trit­to; he was placed at the Real Col­le­gio di Mu­si­ca in Na­ples, and after a publ­ic ex­am re­ceived a schol­ar­ship from Fer­di­nand I, King of the Two Si­ci­lies. He com­posed his first can­ta­ta at age 15, and went on to write sym­pho­nies, or­a­tor­ios, op­er­as, and quar­tets. He event­u­al­ly moved to Lon­don, where he was knight­ed in 1869. In 1871, he be­came “di­rect­or of the mu­sic, com­pos­er, and con­duct­or” at Her Ma­jes­ty’s op­e­ra.