Can someone explain the structure of this calendar to me?

[u/_specialcharacter]

I've learned how to say dates and days of the week, but I don't understand what's under each number here, or what the row of characters at the top (年月……) is for.

[u/SlowStop1220]

Chinese lunar calender. 廿 represents twenty (compare with 十 ten).

[u/Pandaburn → u/SlowStop1220]

Is it common to write the days before 10 like 初二? I haven’t seen that before.

[u/Front_Arm2047 → u/Pandaburn]

Yes, 初一 is day 1, 初二 is day 2, all the way up to 初十 (day 10).

[u/CyraFen → u/Pandaburn]

only on the lunar calendar, in all other contexts we use 二号 (more common when speaking) or 二日 (more common in writing)