Unfortunately there are no set rules for which reading 日 will take, same goes for 人 for にん vs じん, so it really comes down to memorization 
as you keep going, you’ll start getting a “feel” for it though and it’ll come more naturally to you to remember which reading it will be in the different vocab, so try to memorize it the best you can for now!
As for げつ vs がつ for 月 when referring to months, there aren’t really any set rules either but the way I like to think about it is:
- The がつ reading can be seen when referring to something specific, like months of the year
→ “January” → 月 = いちがつ
→ “February” → 二月 = にがつ
→ “March” → 三月 = さんがつ
→ “April” → 四月 = しがつ
etc…
As for something like 何月 (なんがつ) - “what month”, it’s asking for a specific month as an answer, thus this also uses the がつ reading
- げつ can be seen for broader terms, examples being:
→ “end of the month” → 月末 = げつまつ
→ “every month” → 毎月 = まいげつ
–>“next month” → 翌月 = よくげつ
–>“month after next” → 再来月 = さらいげつ
Since these aren’t referring to a specific month and are being used in a more general sense, they are seen to take the げつ reading
Of course, this isn’t perfect, there really aren’t set rules for these – they’re are just patterns I picked up that helped me remember some of the readings, so I hope they can be of use to you too! Keep pushing on and you’ll get better at remembering the different readings for each individual item and it’ll become more second nature to you 
頑張ってね ! 
