Why did God tell Moses to make a bronze serpent on a stick? In Numbers 21:4-9, Moses makes a bronze serpent on a stick to heal the Hebrews' snake bites, as per God's instructions. They don't worship it, but they are to look at it in order to live. Isn't this sort of like an idol, or a graven image? Or could it be a reference to something learned from Egypt, like possible Hermetic disciplines?  

Also, is it related to the image of a serpent coiled around a staff, seen commonly today in the medical field?
I think in order to answer this question, a short detour needs to be taken to John's Gospel, because in it Jesus Christ himself refers to this incident with Moses and the bronze serpent.  In John 3:14-15, Jesus tells Nicodemus, "And as Moses lifted up the serpent in the wilderness, so must the Son of Man be lifted up, that whoever believes in him may have eternal life."

Then, Jesus continues by noting that out of love God sent His Son into the world to save it, not to condemn it (John 3:16-17).  Jesus then says, though, that the world was already condemned: "Whoever believes in him [i.e. the Son, Jesus Christ] is not condemned, but whoever does not believe is condemned already, because he has not believed in the name of the only Son of God" (John 3:18).

The point in Jesus' discussion with Nicodemus is that he came into the world to save a world that was already condemned, and that this salvation comes through him being "lifted up."  The world, and us in it, are "condemned already" due to the sin of Adam and Eve, and Jesus saves us through his death on the cross (i.e. he was "lifted up").  All who look on Christ in faith receive this redemption from death; those who do not are not condemned by Christ, because they were already condemned due to sin.

So, getting back to Moses and the bronze serpent in the book of Numbers.  To put some context around it, the people of Israel had been in slavery in Egypt.  Through Moses, the Lord has delivered them up out of slavery, redeeming them from their captivity.  He is now leading them to the land He has promised them.  Yet, the people continually rebel against the Lord.  They finally "spoke against God and against Moses," accusing them of bringing the people out to the wilderness to die.  Then, they call the manna which the Lord had been sending them "worthless."  Thus, they despise the Lord's grace and mercy in delivering them out of captivity (cf. Numbers 21:4-5).

Therefore, the Lord gives them what their sins deserve: death.  He sends "fiery serpents" to them, which bite the people and kill them.  Sin came into the world through the serpent, Satan, and now here in Numbers the people are suffering the consequences of sin, namely death; they are "condemned already" (Numbers 21:6-7).

However, the Lord provides a means to save the people.  He has Moses make a bronze serpent, the symbol of death, and raise it up on a pole.  All who look on it will live (Numbers 21:8-9).

Jesus connects this event with his own crucifixion.  He is raised up on the cross to die for our sins.  He dies on the symbol of death, apparently "bitten" by the serpent Satan, dying the death we deserve.  Yet, through this symbol of death we have life; everyone who looks on the cross of Christ in faith will live, even though we have been bitten by the serpent and are therefore sinners (cf. Numbers 21:9).

That's a lot of words to say basically that Christ interprets the event in Numbers in light of what he came to do; he came to die by being raised up on the cross in order to save people who were "condemned already" by the serpent Satan and the sin he brought into the world.  So, what happens in Numbers points forward to what Christ was coming to do on the cross for all people.

Indeed, Christ came to save not only us, but also his entire creation.  He died and rose to save a world that was "condemned already," restoring it to the perfection in which he had originally created it (cf. John 1:1ff). This will come on the "Last Day" with his return and the resurrection.