Hi Nikki. I'm going to give you instructions on how to make something that will help make this easier to understand.
Draw a circle on a piece of paper. In the middle of it, write "Verb." On the left of the circle, draw an arrow pointing left. Above that arrow write, "Who is it that..." and "What is it that...." On the right of the circle, draw an arrow pointing right and put "what? whom?" Draw another arrow that says, "to what? to whom?"
Now, write your sentence on a piece of paper and tear it up. Start with your verb. The verb is USUALLY the action in the sentence. In this case, the verb is "are going." Put "are going" into the circle that says "verb."
Now, read your sentence with your left arrow "Who is it that ... are going?" So, basically, "Who are going?" You'll see that it's either "My best friend and I" or "My best friend and me." Here's the easiest way to tell which one it is with this material:
If it answers who is doing the action, it is the subject. ("Who is it that..." or "What is it that...") In this case, the question is "Who is it that are going?" Answer: My best friend and (I/me).
For subjects, we use the following pronouns:
He, She It
Since "I" is listed in there, it belongs there.
If it answers one of the other 2 questions, it would use the pronouns:
Him, Her, It
Since this one answers the question "Who are going?", it is a subject. So "My best friend and I...." is the correct answer.