Warm up your vocal chords by singing scales, arpeggios and enunciation exercises.
Ask every participating singer to look over his specific part (this can be any combination of soprano, alto, tenor and bass along with a few more in between vocal parts).
Ask every singer to have her vocal part memorized as individuals before attempting to sing in harmony. This is especially vital if you don't have sheet music for periodic reference.
Allow each person to sing their part individually along with the accompaniment of a piano or background recording. This will help not only the person singing at the time to memorize their part, but allow everyone else to hear what each part should sound like by itself without harmonization.
Key and sustain the lower-most note of the harmonization on the piano in tandem with the corresponding performer joining in, allowing each higher part to join until everyone is singing their opening note. Begin singing in harmony once everyone is successfully singing their appropriate pitch.