Warm up your vocal chords. Spend several minutes talking in a normal tone, sing along with songs in the most comfortable area of your range or do vocal warm-up exercises at a piano.
Find a range of eight to 10 notes on a tuned piano or keyboard that you are able to sing comfortably. As you play each note, listen intently to the pitch of the instrument tone. Sing a straight, unwavering tone along with it. Gradually do this up and down the note range on the instrument. Do this as a daily singing exercise.
Sing a note first, then play the note on the piano to see how close you are. This will test your pitch finding ability. If you are consistently off the pitch, spend more time doing step 2.
Play one note, then play notes around the held note one at a time to listen for what sounds like harmony and what sounds like a tone clash. Notes side by side will clash. Often, the third interval note above or below is the best harmony. Do this exercise frequently to help you learn to recognize harmony.
Practice with your voice. Play a note on the piano or have a singer hold out a nice well-pitched tone. Decide if you are more comfortable singing higher or lower than that tone.
Experiment singing notes higher than the held note to find what complements it if you prefer a higher pitch. Or experiment with lower notes if you are more comfortable singing lower than the held note. Generally, you should not have to sing too far away from the held note to find a tone that harmonizes.