Months in general, and moons: げつ (今月, 来月, 月末, 月曜日)
Specific month: がつ (四月, 五月, 何月 - see that this is typically a suffix, although the relationship doesn’t work the other way around)
And rarely there are exceptions like 年月日, 毎月 (まいつき is common instead of まいげつ; 月末 can be read as つきずえ)
And maybe other oddities but this is a pretty solid pattern since がつ actually is a thing.
There isn’t as easy to see a one for the other ones you gave though, imo. You can search jitsu/nichi using the magnifying glass icon for other people’s ideas. The more experience you have, the more intuitive it will feel? There are patterns based on meaning and the different period the reading of the kanji was introduced (https://en.wikipedia.org/wiki/Kanji#On’yomi_(Sino-Japanese_reading)) - while there are plenty of exceptions, some things just start to make sense in different contexts more than others. Realistically you will just passively memorise this stuff and get better at guessing.
I’m struggling to word this, hence the vague description. Because it is not hard and fast. But some readings are more common with a literal or specific day (which is tied closely to the concept of a sun), and some readings are more common with the idea or concept of a type of day. Some can appear as both and I’m sure you could argue there are plenty where this doesn’t apply. But it might help. You should develop your own feeling, and realise there are always going to be words where you can’t develop rules for, and also sometimes it’s futile to develop meaning-based rules (imagine, for example, trying to make perfect rules like this for cognates in English when sometimes it’s just that we borrowed from French instead of German or Greek instead of Latin).
The more you practise new words and see them, you will just get used to it. So that is one thing - language isn’t just pure memorisation but also usage. You have a feel for what sounds right and how to spell right in all languages you write properly, even if it’s not perfectly honed.