If you like the Dixie chicks, and can hear the harmonies, transcribe as many of their songs as you can. Similarly with other groups. Once you hit a critical number, it will start to come naturally.
If you like the Beatles, with all their three part harmonies, there are a few books with note-by-note transcriptions. And this site, if you do better by ear, is nothing short of wonderful: How to Sing This Boy Vocal Beatles Vocal Harmony Cover - Galeazzo Frudua - YouTube