If a couple is married in Israel through the rabbinate and subsequently moves to the United States and decides to divorce, what is the requirement for getting a civil divorce as well as going to a Beit Din for a get?

Although the halacha does not require a civil divorce to break the bond of Nissuin (marriage), in the United States one will require a civil divorce to remarry legally (this may vary from state to state as some states allow clergy to register marriage and divorce). This would be true for a couple who married in Israel as well. If they attained a kosher Gett in the US and then returned to Israel, with proper documentation that would be sufficient to remarry in Israel.