Barber shop singers have a method for learning harmonies that is both fun and functional, and seems to readily transfer to other vocal idioms. The guiding principle is simple - find a note that "works", stay there until it doesn't "work" anymore, then find another that "works". I think of it as a kind of vocal tablature. The notes that "work" are 'most always components of whatever chord is being played. Couple this with the many fine tips shared above and you'll be on your way to having fun.