calethyia is right. It depends on what grammatical role the phrase is playing in the sentenfce. The following sentences all follow grammar rules:
Roger and I went to the store yesterday.
The king and I met at a fancy ball.
Your mother seems to like Roger and me.
Big hats look good on the king and me.
In the first two sentences, the phrases in question are all subjects, and "I" is the pronoun that grammarians tell you to use in subjects. In the second two sentences, "me" is the correct choice because grammarians tell you to use "me" in object phrases.
HOWEVER:
In phrases like this, people have been using either pronoun for hundreds of years. In fact, that was the reason that grammarians felt the need to create the rule. So just because the rules tell you to do something doesn't mean that you'll be speaking improper English if you don't do it. In fact, I suspect that most people you know use both of these all the time without knowing it.
The strategy they suggested of removing the first thing from the phrase is a good one, since the same variation doesn't seem to exist when just one pronoun is being used.