It depends on what function the phrase is playing in the sentence.
If the two of you are the subject of the sentence, you would use "Bob and I," for example:
Bob and I are going to the movies.
If you're the direct object, you'd use "Bob and me" or "me and Bob":
The committee greeted Bob and me.
The easiest way to know for yourself is to pretend Bob is elsewhere, and think which word would be used just to refer to you. You wouldn't say, for instance, "the committee greeted I," nor would you say "Me am going to the movies."