What was Moses' staff made of? Copper? What's relationship between Moses' staff being turned into a snake in Parsha Sh'mot and the incident in Parsha Chuckat if there is any?
It was made out of Sapphire and had the words דצ"ך עד"ש באח"ב , (an acronym of the Ten Plagues) inscribed on it. See Pirkei Avos chapter 5:6 with its commentaries.There are also midrashim on this topic.
From Pirkei D'Reb Eliezer 40
This is the history of the staff: In short, Moshe was the only one who could pull the staff from it's place which showed he will be the redeemer of Israel. Also it was created Ben Hashmashos on the 6th day of creation and was given to Adam who passed it down until Moshe came upon it.