The nuanced differences between は and が seem to be one of the biggest hurdles of new learners. An important part of learning the difference is to learn the difference between the Japanese grammar concepts of sentence subject and sentence topic.

Basically, the "topic" of a sentence is more generalized, whereas the "subject" is more specific. The topic is something that is already known or can be assumed and therefore can be omitted given appropriate context, but the subject cannot. Frequently, the topic and the subject are one in the same, in terms of the more broad concept of grammatical "subject", but the key difference is in how a sentence relies on its subject. The particle は is the topic marker, whereas が is the subject marker. I'll use the example sentence you gave to demonstrate some other sentence types.

これは田中さんの本ですか？ "Is this Tanaka-san's book?" - Here, the topic is これ, "this". "(As for) this, is (it) Tanaka-san's book?" As the topic here is generalized, and may be clear without actually enunciating it (e.g., if someone is gesturing to a book), the topic can be omitted entirely. Thus, これは田中さんの本ですか？ and 田中さんの本ですか？ are functionally the same, provided that the context remains the same. In English, there are lots of instances of generalized subjects - personal pronouns, it, this, that, etc. Many of these serve the purpose of taking the place of a more complex subject to avoid redundancy. In Japanese, they avoid this redundancy by declaring a topic and then omitting it completely in subsequent sentences once all speakers are on the same wavelength.

どれ(の)本が田中さんの本ですか？ "Which book (of these books) is Tanaka-san's book?" Here, the subject is どれの本 "which, or a, book (of a particular set of books)". Because we're asking a specific question about a specific set of things that must be specified in order to understand the meaning of the question, we cannot omit it - it is the subject of the sentence, and is the crux of the question itself. (The response to this sentence would be この本が田中さんの本です.)

田中さんの本はどれの本ですか？ "Which book is Tanaka-san's book?" Here, the topic is 田中さんの本, "Tanaka-san's book". You're asking the question about his book, the question is the same but the emphasis is different - your inquiry is now about "Tanaka-san's book" rather than "the set of books which you suspect one of belongs to Tanaka-san". The actual question itself comes after the topic, so it's は instead of が. "(As for) Tanaka-san's book, which book is (it)?"

Try to think of は as the "subject" of the broader conversation, and が as the subject of a particular clause or sentence. As it would be cumbersome to have to repeat the topic every single sentence, it can be omitted, provided that everyone involved understands the context; when a sentence in a conversation suddenly takes on a new, specific subject just for that sentence, but isn't going to steer the overall conversation into a new topic, が would be used instead.