Play a major scale on the piano and sing along using the words Do, Re, Mi, Fa, Sol, La and Ti until you are familiar with the sound of a major scale. Then stop playing and sing the scale on your own. Now play the scale again to check and see if you sang it correctly. Once you can sing the scale correctly on your own, try playing another note on the piano and sing a major scale starting at that note. Again, check to see if you sang the notes correctly. Once you can sing a major scale starting on each note, begin again and use the same process to learn the minor scales.
Once you have trained your ears to recognize scales, it is important to learn intervals. Play the first note of a scale. Now play the second note in the scale. Sing this interval on your own without playing it. Now play and sing the first note of the scale followed by the third note. Sing it on your own. Continue to do this with each note in the scale. After learning each of these intervals, test your memory of what you have just learned. Play the first note of the scale. Now, choose which interval you are going to sing and sing the note in your head. After singing it in your head, play it on the piano to check your accuracy. Repeat this process every day until you can sing any interval in your head correctly. This may take years of practice to master.
It is important to learn each voice part in the songs you are playing. Interval ear training will come in handy when learning songs because each voice part in the song is made up of notes going to different consecutive intervals. Because songs are longer, it will help to record yourself singing. Play each voice part on the piano. Then record yourself singing each voice part. Now play back the recording you made while playing the song on the piano at the same time to check your accuracy. Repeat this until you have learned and can sing each voice part correctly.