Which is correct between Cathy and I or between Cathy and me?

Between "Cathy and me" is correct. "Between" is a preposition, and you use "me" as the object of a preposition. Always try it with another proposition, such as "for," and make it singular. Would you say "for I"? No, you would say "for me." This applies to more than one person, so the proper usage is "for you and me," the same as "for me." This applies to "between" as well.