In real life, using past perfect in the first clause would be optional. Past perfect is mostly useful or necessary for making a story clearer when you explain the events backwards, which you are not doing in your sentence. However, in an exam situation, you should never miss the opportunity to increase the range of tenses that you are using. So, in that case, you should use past perfect, even though it's not necessary or very common most of the time.

There is no need to use past perfect here because the order of events is made clear by 'after' and by the order in which the actions are said - learned and then began. In English, we assume that simple actions (past simple, present simple, future simple) happen in the order they are said in the sentence. This is the case here so past perfect is not necessary.

The past perfect tense is used to place an action before another action in the past. In other words, if you consider two actions that both happened in the past, the one that happened first will be in the past perfect tense while the one that happened after will be in the simple past tense. So I believe that if you write your sentence using the past perfect: "After he had learned about the benefits of regular sleep, he began going to bed early", it's correct. I hope that helps.