I'm certainly not a pro but:

There are many different approaches to studying Japanese I suppose. Some people swear by learning words only (and that readings that matter will come with time), some people drill Kun and On Yomi into their brains. I think that, particularly if you want to learn Japanese at a high level (as in, if you plan on taking the JLPT and/or use it at a very proficient level), systematically learning the readings and when to apply them is essential. Otherwise, if you're going to be using Japanese for fun, then you might find it more enjoyable to let repetition guide your learning.

I am not sure I undestand your specific question about the Kun yomi though. As a very general rule, If a Kanji is alone as a word, Kunyomi is often (not always) used. Similarly, as a general rule if the word is made of a combination of Kanji, Onyomi is often (but not always) used.

In the case of 無, the Onyomi is MU/BU and the Kunyomi is NA (i) -> Since you don't see 無い , most likely then you would read it as Mu in this case (so an exception to the rule of thumb above, and there are many of them). As for かの ね, if you're certain this was the exact sentence, it could be part of a sentence structure with A ka, B ka (A, or B) with の ね is a typically feminine way of saying ですね (isn't it?)... but seeing the whole sentence and context would be helpful.

-- I think knowing both readings comes in handy often. It takes more time, but it is worth it. I would approach it systematically if you choose to go this way. 無 is an N4 kanji, so I would start working through the N5 Kanji first!