Hi, I got the same problem and become frustrated at the issue when to use にち/じつ/ひ. I would say they are equally common and you just have to learn them as they appear. Type 日 in the upper right corner of WK and you’ll get all the words that use 日 across all levels.
The same thing happens when it comes to 月: げつ, がつ, つき are just random and sometimes shortened to just one syllable or connected to the next syllable by small tsu.
as for 月, i figured out that inquiring specific months are がつ while inquiring general months are げつ. i.e. if the usage fits as a general case for whatever month versus if it fits for a specific month.
don’t know if this breaks in higher levels.
This may or may not be helpful, but I’m usually able to figure it out based on what ‘sounds the best’. It will typically be にち unless that sounds odd, in which case it will be じつ. For example in 先日, せんにち would be awkward to say because of the double n sounds. せんじつ flows much better.
I haven’t got anything as clear as 月 being specific months, but I’m remembering it as じつ when it doubt, but 毎 and 何 tend to use the stranger readings - にち, つき and とし. (then again, 何年 is なんねん so…)
Still though, something more concrete would make remembering easier.
I think there is a pattern, because things that ‘sound the best’ are usually based on an underlying linguistic phenomenon. Couldn’t tell you what it is in this case though, if it exists I think it’s too complex to help you. edit: this part is wrong, check out @Saida’s reply below!
However, most two character pairings with 日 as the second character are pronounced as じつ (at least in WaniKani). So you could try just memorizing the exceptions. 毎日、何日 are the only ones I can think of.
毎日 まいにち Every day
何日 なんにち How many days/which day of the month
All of these can mean both a timespan of the number of days, or the specific day of the month. Except ついたち, that one is specifically the first day of the month.
So, the pattern is, most days end in にち, except the exceptions.
Also, all words that refer to Japan the country, (and have the 日 kanji in it) will be にち, like 来日、駐日、日欧、日(ち)本.
Days of the week end in ~曜日 ようび
Then, in compounds, if it is the second kanji, it will most likely be じつ, unless it’s from the above list.
Sometimes it will be び, like in 記念日、誕生日, which are compounds with the suffix ~び (day of remembrance, day of birth).
The explanation of “a specific month” is arguably more correct, though; number + 月 can also be using the 月 counter. For example, I believe that the 二月 in 二月に一回 would use ふたつき if you mean “bimonthly” or “once every two months”, but にがつ if you mean “once in February”.
何月 can also be read as なんつき if you mean “some month” or “whatever month”, but なんがつ if you mean “which calendar month”.
I simply remember that all of the months that I know in English have the letter a in them:
January
February
March April
May
I believe there are a few more months, but I’m sure the pattern continues.