On’yomi is the imported Chinese reading of a kanji, which came with the kanji when they were imported from China. Kun’yomi is the native Japanese reading, from where kanji were applied to Japanese words that already existed.

General rule of thumb is that you use kun’yomi when a kanji’s standing on its own or has tacked-on okurigana, and on’yomi when it’s in a compound word with other kanji, but there’s about a thousand and one exceptions.

For example, 話 uses its kun’yomi はなし when it’s standing alone. With okurigana, as 話す, it’s read as just はな. In a multi-kanji word like 電話, it uses its on’yomi, わ (the word is read as でんわ).

An exception: 川口, despite being a multi-kanji word, uses kun’yomi for both kanji (it’s かわぐち). I think in general, the more concrete and natural-world-themed a word is, the more likely it’ll be to use kun’yomi. 川口 = river mouth.

Your best bet is to just start learning vocab, and you’ll start to get a feel for what readings to use where over time.

You really can’t avoid learning kanji altogether. It’s like learning the alphabet in English but never learning any words.

Are we talking about the anki deck? I never paid much attention to the anki decks, but I think he’s specifically testing the on’yomi, there. Honestly, I don’t think there’s much benefit to be gained by learning the readings in isolation, because there’s nothing to anchor it back to regular usage. Like I said, learn vocab, and you’ll learn the readings as a result.

For example, suppose you learn 能力 (のうりょく) = ability. Then you encounter 協力 = cooperation. By comparison to 能力 which you learnt before, you might infer that the 力 is read as りょく – and you’d be correct (it’s きょうりょく).