A metronome is invaluable to give a student a sense of an absolutely regular pulse. In my experience most students do not have that, and when they hear themselves playing, they often think they are regular when they are anything but.
So I introduce the metronome in the very first lesson to do rhythm exercises (get a simple piece, ignore the pitches, and simply play the rhythm right on the beat with the metronome). This will go on for a while, and once the student gets the knack of it, I immediately discontinue it and only ever use the metronome again if this sense of regular pulse is lost.
On rare occasions I will use the metronome to figure out difficult rhythms, but again I will discontinue it as soon as a feeling for the rhythm is achieved. Ultimately no piece should be played metronomically, and a metronome is really a crutch that can be very helpful for a period of time – if and only if you cannot use your legs – but that should be abandoned as soon as you can walk by yourself.
I rarely if at all use the metronome to develop speed (start at a slow beat and repeat going up a notch every repetition) because I believe it creates speed walls. I will however do the opposite: use a slow metronome to avoid rushing up (it is far easier in general to play fast than it is to keep a steady slow pace).
Finally – and in my opinion – the area the metronome is the most useful is in mental practice: By going through a score with the metronome on, you ensure that you are performing the piece in your mind in real time. The mind tends to fast forward and skip whole sections unconsciously, so it is very easy to deceive oneself that one has gone through the piece mentally at the real tempo, when in fact nothing of the sort has ever happened.
I hope this helps.
Best wishes,
Bernhard.