Start by listening to music--really listening. Classical music, while not everyone's favorite, is the best style for ear training. Try to learn the sounds of different instruments and try to pick out their parts in the piece. Learning to distinguish qualities of sound, such as pitch and tone is the first step.
Here is the simplest exercise for discerning pitch: Have someone sit at a piano, or keyboard and play two notes in succession, going either up or down the keyboard. Have the trainee listen, eyes closed, to the notes. Let them say whether they think the second note was higher, or lower. Begin with notes that are separated by two steps or more, then notes a step apart and finally half steps. Work at this for no more than 15 or 20 minutes a day. After three or four sessions, you will find that the trainee will get it right immediately almost every time.
Once the student has become proficient at determining whether a note is higher or lower, play a single note from the octave beginning with middle C. These notes will typically be in range for most voices and easy to hear. Have the trainee sound the note back with their own voice. Repeat the process with each note keeping track of which ones, if any, they were successful in matching on the first or second try.
Using these notes, play three tone scales beginning with the correct matches and ask the trainee to sound them back in their own voice. Baby steps are essential, because they give a sense of accomplishment. They may sing two notes correct the first time, but soon it will be all three.
Continue to train 15 to 20 minutes a day. Make sure it is concentrated time. Do not exceed that until the trainee has mastered the art of matching pitch fairly consistently. If there is one note that is especially clear, find a song that repeats that note frequently with a simple tune in their range.
If you cannot find a song that repeats that note, simply find a song that repeats any note and use a notation software, such as Band in A Box, to transpose the piece to make the repeated note that natural tone that they sound well. Teach them the song and have them practice until they can sound all of the notes fairly consistently.