There's confusion when it comes to the difference between me and I. For example, we say Not I, but Woe is me; John and I went to the shop, but Sam gave John and me a book. Which pronoun do we use, and when?

Let's start with the easiest first. In the last example, Sam gave John and me a book, the pronoun me takes the dative case, meaning that an action is given to someone or something. We use me for the objective case as well: Sam took John and me to the bookstore. Again the subject is Sam, not me. If all this is too damn complicated, I have a tip: take out the unnecessary character and see what makes sense. Sam took I to the store or Sam took me to the store. This trick also works with the third example. John and I went to the shop becomes I went to the shop.

Who killed Cock Robin? It is I. In this example, it is is a transitive, or linking, verb. No action occurs, and an old rule is that subject pronouns follow these verbs (I, he, we). This also applies to the negative answer: not I. Of course, in modern colloquial usage, few speak like this, and you'd be better off being wrong and fitting in than being right and sounding like a Shakespearean nobleman: Who ate my waffles? Not me. Wasn't me. T'was I, Lord of Grammar and Judgment. 

That's a line from Radiohead's "Creep:" I wish I was special, you're so fucking special. Of course, the only reason I'm using this line is because its grammar is wrong. Radiohead should have listened to If I were a rich man. Notice the difference? It's that glaring counterfactual conditional, or contrary-to-fact conditional statement, of the past subjunctive clause. 

Subjunctive mood is something I learned from taking too many years of Latin and the occasional Prairie Home Companion show. In English we use this mood to describe things that are imaginary or conditional. Since that probably didn't help explain, here are a few examples:
I suggest that you leave. You haven't left, thus that you leave is in the present subjunctive mood.
I wish I were a woman. I am not a woman, therefore I were is past subjunctive.
If I were to have a hammer. I don't have a hammer, and were to have is more past tense than if I had, so it's pluperfect subjunctive.

The difference between who and whom is so simple that we don't really need to know the grammar behind it, but I'm going to explain the grammar anyway, cause then you'll know why to write who or whom. Who belongs to nouns and pronouns of the subjective case, meaning they are the subject of the sentence. Here are the subjective pronouns: I, you, he, she, it, we, you (plural), they. With these, you shall use who. Pronouns or nouns that act as objects are in the objective case, and the following is the list of objective pronouns: Me, you, him, her, it, us, you (plural), them. These use whom.

There's a trick to figuring these out: remove who or whom and replace it with one of the above pronouns. Let's see some examples:
Incorrect: Who can I trust? (I can trust he? No. I can trust him? Yes.)
Correct: Whom can I trust?