My students tell me remembering dates in Japanese is a nightmare. Yes, I understand. But don’t worry, even if you say ろくがつにじゅうにち, people will understand what you are talking about. It’s not a life or death problem!!
Having said that, 二十日 being called はつか, instead of にじゅうにち, is because "twenty" is a special number for Japanese.
One of the special meaning is that the legal age in Japan is 20. "20 years old" is written as 二十歳 as usual but pronounced はたち (hatachi), not にじゅっさい (nijussai).
Be aware if you are not quite 20, you are not legally allowed to consume alcohol in Japan!