It's been something I've been thinking about for a while, but could never answer in a more concrete way. This is most apparent when I compare the functions of 何 and どう. I know that 何, どう and どんな are great basic building blocks, but I'm not sure about what each implies when used to ask a question. Does anyone know?
I assume you are already familiar with the "1-to-1 mappings" that @Jamal posted above. That's not all there is to it, though.
You can think of it as limits to areas that each question may address, if that helps. For example, you can say 「何をすればいい？」 or 「どうすればいい？」, as their target areas overlap, but the second is, to my knowledge, a more common and natural construction for "What should I do?" For an emphatic statement, you can say 「何をどうすればいいんだ？」
Essentially, it doesn't matter what the apparent 1-to-1 mappings to English may be, because the target area of a question word is more based on context, construction, and practice. Another example of overlap that breaks the 1-to-1 mapping is どういう and 何, i.e., 「どういうわけか、彼は日々に変わっていった」 "For some reason, he changed as the days went by" is equivalent to 「理由は何かわからないが、彼は日々に変わっていった」 "I don't know the reason, but...", but the second is definitely a clunky phrasing that would not pass muster in a submitted assignment.
I tend to think of （こ・そ・あ・ど）んな as contractions of （ど）のような〜, as it emphasizes the question of aspect in my mind. よう is occasionally written in kanji as 様, which strengthens the association with appearance or manner. It's a stilted translation, but "What manner of [thing]?" is a good way to highlight its wide range. 「こんなことで私達が別れるなんて…」 "[I never thought that] we would break up over something like this...". The "like this" part is essentially contained within the two words この[this] and よう[like]. 「そのとき、容疑者はどのような動きを？」"At that point, what manner of movement did the suspect perform?"/"At that moment, exactly how did the suspect respond?"
There are two ways to learn something like this that I know of. You can either get a Japanese grammar dictionary or collocation/phrase book, or read a heck of a lot of books/manga and watch TV. If you give it a solid year of heavy consumption you should start to feel unconscious promptings about what is "right" in one situation or another.
You'll see どんな in the sentences that ask: "What kind of watch is that?" or "What the heck is that thing?".
You'll see 何 in sentences that ask: "What is he doing?" or "What is that thing?"
Note: In the first example, the speaker is expressing surprise, disgust, or heightened curiosity. In the second example, the speaker is just asking a straightforward question, ie. a plain inquiry.
You'll see どう in sentences like: "How's the food?" or "How is he faring?" or "What's his prognosis?".
Note: "What's his prognosis?" can be reworded as "How is his chances?"
I hope this helps with your understanding of the 3 words.