If I may chime in,
In order to have harmony two or more different notes must be sounded simultaneously. The resulting sound should be "pleasing." To some extent "pleasing" is in the ear of the beholder. What sounds pleasing to a hearer is more or less what the hearer is used to hearing. What sounds pleasing to a Chinese ear might not be what is pleasing to a European ear.
There are basically two way to go about creating harmonies: beforehand and on-the-fly.
If you want to create harmonies beforehand, you can take classes on harmony at your local school of music to learn the theory. Then you sit down with a tune, a piano, and staff paper and create a harmony for that tune. You learn the harmony and sing or play it as needed.
If you want to create harmonies on-the-fly, you need to a little theory, but you especially need to immerse yourself in the harmonies you want to create. Listen, listen, listen. Then listen some more. In my experience, creating harmonies on the fly is not easy for the beginner and the ability is not quickly acquired. For traditional American harmonies it helps to grow up singing in a choir (Southern Baptist if possible).
There are a always a few harmony classes taught during vocal week at Augusta (www.augustaheritage.com)