If you are asking someone to come along with you to the theatre, you'd say "Will you come with Amy and me."
There are several ways to tell which is correct. One of the most obvious is to remove the other party entirely. In this case, using the same scenario, you'd say: "I went to the theatre." You would not say, "Me went to the theatre."
Likewise, using the other example, you would not say: "Will you come with I." Therefore, in that sentence "me" is correct.
Hope this helps, a bit! It is one of those things which you have to get into the habit of running through your brain, before engaging the output.