Listen to music and note the difference between the melody and the harmony, if there is one. The melody comprises the notes that are most prominent in the piece. Usually a singer sings them or the lead instrument plays them. The harmony is the notes that accompany the melody to give the music a fuller, richer sound.
Stay away from the melody if harmonizing is hard for you. The melody frequently gets stuck in your head. If you haven't learned to harmonize yet, that can make it tough to find the notes above or below.
Find the harmony notes. This is sometimes best done with a good set of headphones on so that you can hear all the sounds. Background singers frequently sing harmony to the lead singer's melody. Note how the harmony follows the melody in steps up and down. If the melody skips from C to G, the harmony also moves four notes up or down.
Start singing when you've found the harmony and practice to get it stuck in your head. The goal is to sing the harmony as confidently as the melody, which means you'd know your starting note and start singing without searching for the right note. This requires a lot of practice and sometimes a lot of trial and error.