Learn basic scales before attempting to sing harmony. Learn the lead part so that you can learn the harmony easier.
Try playing the harmony on an instrument such as the piano to help you hear the notes you will be singing. Play or sing the scale of the harmony to accompany the melody before attempting to sing the notes out of scale order.
Listen to a teacher or knowledgeable person who knows the harmony of a particular piece. Train your ear to recognize the melody and the notes that compliment the main theme.
Train your ear to hear dissonance and resonance within the music. When a note is sung correctly in harmony, there will be a vibration that accompanies the sound. Note that if it sounds right it most likely is right; discord in harmony is fairly easy to detect once the ear is trained.
Attempt to sing the same notes as the lead part and then sing the notes in the chord above or below the lead part. Know that the baritone part is typically the most difficult and often has a strange succession of chords that will not sound like a tune in its own right.