When writing an essay or short paper for school, should I state my intentions and outline the essay's structure in the introduction? Or should I let the essay speak for itself?
I don't think anyone can give you a simple answer to that question. There are many ways to structure an essay.
Some people like to begin by saying "here's where I'm going". This can be helpful if the essay is long and complex, or if the subject is not obvious from the beginning. Like, if you're writing an essay about your views on how to reduce unemployment, and you start out talking about the Franco-Prussian War because you're building up to some analogy with how problems during that war were solved, it might be a good idea to give the reader a clue up front about where you're going with this.
But other times it is better to just launch into your discussion. For example, to start out by saying, "First I will discuss the problem and then I will give my proposed solution" is likely to be superfluous. The reader will easily see that anyway.
Many writers like to begin an essay with a hopefully-interesting anecdote. For example, if the writer is discussing some large-scale national problem -- economics or politics or foreign relations or whatever -- sometimes he will start with a little story about how one particular person was affected by this problem, to make the whole subject less abstract and maybe get the reader's interest.
Another possibility is to begin with a literary allusion or famous quote. Like find a quote from Shakespeare that ties into your point.
Etc. There are many, many ways to get started.