The best way that I've found (or at least the one that works for me and my students) is to make the harmony the melody. This is more difficult when you are trying to harmonize to a song that you already know the melody really well to, but it's still possible. You have to sing just the harmony over and over until it becomes your melody. Then add the original melody line and try to hold on to your melody (harmony). Plugging one of your ears to limit the input of sound from the other part is also helpful in helping you stay on track with singing your melody.
It's definitely something that gets better with time. Some are born with better ears for this than others, but it is a skill that can be developed. Before trying to hold a harmony on your own, sing in a choir where there is an entire section of singers singing your line; that's the best practice.
There are many tips on how to harmonize. The one that seemed to stick out in my mind would be the idea of singing along with the song. Hearing a recording of the song that you can match your talents to and harmonize along with that will help you better understand how to manipulate your voice into the desired harmony. Another option to better understand where your voice and pitch need to be would be to sing the song in a higher or lower octave, allowing you to understand how to train your voice into the desired harmony. The concept of continually practicing, like with any other endeavor, seems to be where the penchant for successful harmonizing lies.
The words harmony and harmonizing are used in many different ways. In music it refers to a specific quality of music, while in arts and design it refers various aspects of a design or work, of art - such as colour, shape, and size - being in line with each other or supporting the effect of each other. These words are also used to refer to the life of individuals and their relationships with others individuals and the world in general. It appears to me that this question relates to harmony and harmonizing in this last of the three meanings.
It is rather difficult to define or pinpoint exactly what harmony of this type means. But one thing is sure, this harmony is invariably accompanied by absence of conflicts of interest between the individual and the person or the aspect of life with whom harmony is desired. This harmony may be reflected in outward actions, but its essence is in the mind. If a person is in love with something - for example, the environment - he or she is note likely to knowingly act in way that harm it.
So the best way to develop harmony in life is to try and see what is good and useful in others. An attitude of giving to others, rather than selfishly only asking from other is another way to develop harmony. This sounds rather idealistic. But then harmony is an idealistic concept.