For Kunyomi, you pretty much have to learn that word by word (just like Kunyomi suggests 訓読み). There's nothing systematic to learn this from what I've seen, outside of adding words that use them.

For Onyomi (音読み), there's some luck. Not counting rare readings, of the 2041 Kanji in RTK about 90% of them have 1 unique Onyomi per kanji. The other 200 or so kanji have only 2 common readings. That opens the door to learning them systematically either through memory palace methods (Kanji Town, Movie Method) or key primitives (RTK2).  On this forum, it appears memory palace methods are most populary.

To add a bit more perspective: 12 Onyomi cover 500 kanji.; 34 Onyomi cover 1000 Kanji; 78 Onyomi cover 1500 kanji; 171 Onyomi cover 2000 kanji. If you limit what you want to learn systematically to say any onyomi that covers at least five kanji, then you can learn onyomi to 1869 kanji using just 137 different onyomi.

The benefit to knowing onyomi ahead of time comes up in reading new material with new words. You gleam not only meaning, but pronunciation from the kanji itself in the word. Not essential early in your studies, but something to keep in the back of your mind to add in sometime later.