Grammar: Using "I or Me" and "Who or Whom"

I confess, I've broken a few grammatical rules in my time. Sometimes I've done it on purpose, sometimes inadvertently. But, there's one grammatical error that makes my skin crawl. It's using the words "I" and "me" incorrectly. This is a very common error. The sad thing is, sometimes people do this when they're trying hard to be correct and proper.

Many people would say a sentence like this, "They invited Susan and I to a party," and feel it was correct. The word "I" is just a classy feeling word. The word "me" is more like a grungy relative you try to hide. At least, this is how people treat these words. You know when someone uses "I" a lot; their intent is to take the time to speak properly. Well, in this case, "I" is not proper. The right sentence would be "They invited Susan and me to a party." Why is this correct, when it's almost second nature to some to use "I" in a case like this? The word "I" is used as the subject of a sentence and "me" as the object. But, if you get confused, or don't want to mentally check your grammatical rules, there's a way to tell. In a sentence that contains "and I" or "and me" take out whoever else is a part of the sentence. What if Susan wasn't invited to the party? Would you say, "They invited I to a party"? Not without getting some strange looks. A "me" fits perfectly here. And, when is it proper to use "I"? If the sentence was, "Susan and I were invited to a party." Leave poor Susan crying at home, and you would naturally say, "I was invited to a party."