This question is asked at the end of a long comment at the end of my post on the topic, “Why does a serpent represent what Jesus did on the cross?”
God gave Moses the power to turn his staff into a snake as a sign to authenticate his ministry before the Israelites. But I’m not sure that the snake itself represented the authority and power of God, or of Moses as God’s emissary. We find out shortly afterwards in Exodus that this was the kind of sign that Pharaoh’s magicians were also able to do, and when they pitted their arts against Moses, his snake consumed theirs, showing that God’s power was greater. But once again, I don’t think we need to look for symbolism in the snake itself.
I also don’t think there’s necessarily a connection between God giving Moses the power to turn his staff into a snake and God commanding Moses to make a brazen (brass) snake and put it on a pole. The simple purpose of this was to provide a visual focal point for those who wanted to turn from their rebellion against God and trust Him for healing from the poison of the snakebites.
If there’s any connection between the two incidents, it’s that venomous snakes are dangerous and potentially deadly; that’s why the magicians chose to produce them–to make a memorably scary impression on their audience–and that’s why God used them to send a plague among the people.
In other words, at least as I see it, just because there are snakes involved at two different points in Moses’ ministry, there’s not necessarily a symbolic significance to them, or connection between them, beyond their plain role in the narrative.
There are other places in the Bible where snakes do have a symbolic significance, but this is pointed out clearly in the text, for example, in the book of Revelation, “The great dragon was hurled down—that ancient serpent called the devil, or Satan, who leads the whole world astray.”
A good interpretive principle to apply is not to look for symbolic significance in, or attribute it to, an element in narrative unless the text itself points you clearly in that direction.