Practice singing major and minor scales. Knowing these scales up and down will make learning intervals much easier. If you are taking vocal lessons, you are more than likely working with a teacher who uses a piano. If you are attempting to learn without a teacher, use any piano or keyboard yourself and sing along with each note as you play the scale. A major scale is recognizable as the familiar melody, "Doh, Re, Mi, Fa, Sol, La, Ti, (High) Doh." An example of a major scale is C Major. It includes the notes C, D, E, F, G, A, B, C. An example of a minor scale would be A Minor. It includes the notes A, B, C, D, E, F, G, A. The first and last notes of the scales are one octave apart. Remember, it is not important that you be able to sing in a particular key on command. When given the starting note, however, you should be able to sing the rest of the notes as a major or minor scale.
Practice identifying major intervals. After familiarizing yourself with the major and minor scales, you are going to train your ear to recognize various major intervals. There is the Major 2nd, Major 3rd, Perfect 4th, Perfect 5th, Major 6th, Major 7th and Perfect 8th. The name of the interval describes the distance between the notes. For example, a Major 3rd is recognized as the sound of the first note in a major scale followed by the 3rd note in that same scale. No matter what major scale you play, that interval will be the same. The reason certain intervals are titled "Perfect" is because they remain constant in both major and minor scales. It also may help to keep familiar song references in your head when identifying intervals. A Perfect 4th interval, for example, can be recognized as the first two notes of Amazing Grace.
Practice identifying minor intervals. This is the Minor 2nd, Minor 3rd, Perfect 4th, Perfect 5th, Minor 6th, Minor 7th, and Perfect 8th. Keep in mind that because major and minor scales are different, a Major 3rd interval is going to sound different than a Minor 3rd interval. The distance between the first and 3rd note in a major scale is different than in a minor scale. An example of an easily recognizable minor interval is the first two notes of the musical theme in the movie, Jaws. This is a Minor 2nd.
Practice singing intervals. Having familiarized yourself with all the various intervals in both major and minor scales, try singing them yourself. Start by singing any note and then sing the Major 6th, followed by a Minor 3rd and then a Perfect 5th. Have your teacher play the starting note on a piano and then ask you to sing a particular interval for that note. If you do this exercise consistently and frequently, it will only be a matter of time before you can sing intervals on command without even thinking.