Once a student is comfortable with basic chords and maneuvering through various changes, then I have them approach using a pick. I prefer having a focus on one hand at a time before putting them together so that they will gain more confidence in one segment of their playing. For most beginners or hobbyists, it makes more sense to get comfortable with chords first because most of their enthusiasm for guitar will be experienced from being able to change the chords in time with a song on a CD or iPod. There are also many songs that use arpeggiated chords, so the left hand has to be able to get the changes cleanly first anyway. For those kinds of songs, a student can play along with the song by just strumming the chord along with the changes. Once they are good at that, they can approach the actual picking.
The following video demonstrates holding the pick as well as how to move it across the string. Generally speaking, you should try to use as little pick as possible to strike the string and make your pick strokes very small as well. I have my students anchor the heel of the right hand on the bridge and also place either the pinky or ring finger on the body of the guitar for greater stability.