When writing an essay, do I end a paragraph with a question or begin the next paragraph with a question?
Your question suggests that you want to know how to transition smoothly from one paragraph to the next, which is an important concern. However, there is more to this skill than just the placement of questions in paragraphs, so let's go over transition generally.
First we need to make a distinction between the transition from an introduction to the first body paragraph and the transitions from one body paragraph to the next. For the former, we usually rely on the thesis statement at the end of the introduction to provide a smooth transition to the first body paragraph, a way of saying to the reader, "Here are my position and supporting points, and the first point I mention will be the basis of the next paragraph." What we do from one body paragraph to the next is a bit different.
When we are moving from one body paragraph to the next, we need a "bridge" for the reader to go over, from one topic or point to the next topic or point. That bridge need not be a question at all. It can be, but too many questions in an essay can be annoying to the reader, who has questions but probably wants answers! What I like to do is construct a sentence that I place at the end of my body paragraph that acts as my bridge. In that sentence, I mention the topic of the paragraph I am in and the topic of the paragraph that is to come next. It is almost like foreshadowing in a way, preparing the reader for the next idea I'm going to discuss. It could be a question, but it certainly does not have to be a question.
Let me give you an example. Let's suppose I am writing an essay on The Kite Runner (Hosseini) and am discussing the reasons that Amir feels superior to Hassan. I have a body paragraph in which I am discussing Amir and Hassan's ethnicity, and in my next body paragraph, I am going to discuss their religious differences. Here is a sentence I could use at the end of my body paragraph on ethnicity to move to the next paragraph on their religious differences:
While it is clear that the ethnicity of the boys made Amir feel superior, it is also clear that Amir's version of Islam made him feel superior, too.
That is the bridge that will take the reader from the ethnicity paragraph to the religion paragraph.
Some writers have a preference of placing their bridge sentences at the beginning of the new paragraph, and that can be effective, too. Once you have mastered this technique, you can try either way and also can try either way with questions. For example, I could also write this:
While ethnicity contributed to Amir's feeling of superiority, did religion play a role in this as well?
That question could be at the end of the ethnicity paragraph or at the beginning of the religion paragraph. If a question works well, that can be a good variation for a bridge statement. However, again, using many questions is usually not a good technique in writing. Practice using statements and questions and see which is more effective for the essay at hand. Using this formula in writing will serve you well as it becomes more a more natural part of what you do, more like an instinct after you've been doing for it for a while.