Why does Nietzsche think that the belief in a soul is a "superstition"?
Nietzsche identifies many of the errors in traditional philosophy with a misunderstanding of and a heavy reliance on grammar. One of his pet peeves is the misunderstanding of the subject-predicate form. Because we can separate sentences like "I think" into a subject and a predicate, we come to see the subject and predicate as distinct. There is the subject, "I," and then the act of thinking is a predicate tacked onto it so that the "I" is some entity distinct from the thinking itself. If we were to ask what this "I" is, we might say that "I am a thing that thinks" or "I am a human being," but in each case, we would be attaching a predicate onto the "I," a predicate that can also be detached. "...am a human being" is just one thing we can say about "I," but it doesn't tell us what the "I" itself is. Ultimately, there is nothing that is inseparably attached to this "I," and, according to Nietzsche, we come to name this nothing a "soul."
What does Nietzsche take to be wrong with our understanding of cause and effect?
Nietzsche complains that we "reify" the notions of cause and effect, that we come to see them as things. Our use of language leads us to believe that there is something in, say, a billiard ball--some sort of "causal power"--that allows it to "effect" the movement of another billiard ball it comes into contact with. Nietzsche suggests that there is no such thing as cause and effect in nature: they are just concepts that we invent and apply to the things we observe. Interestingly, this position is almost identical to ##Hume##'s, a philosopher whom Nietzsche condemns as being "a debasement and lowering of the value of the concept of 'philosophy'" in section 252.
Describe Nietzsche's "experimental method." How does it relate to free spirits and the "philosophy of the future"?
Nietzsche's experimental method is not the one used by scientists. Rather, it represents a willingness to look at a matter from any point of view, and to follow out that point of view to all its consequences. For instance, many of us will find Nietzsche's assertion that all human interactions ultimately consist of a struggle for power quite disagreeable. Perhaps Nietzsche feels the same way, but his experimental method dictates that he cannot discard that suggestion simply because it is unpleasant. He must try it out, see if it works, see if it is a good interpretation of the world. This ability to look at matters from any point of view is crucial to the flexibility of mind that Nietzsche values in a free spirit, and is necessary to the "philosophy of the future" if it is to have the courage to uncover assumptions and prejudices that have glued down philosophy for millennia.