For me, the best “documentation” is to go to
Admin -> Badges
and look at the query used.
Of course being familiar with tables, table fields, and SQL query syntax will help you to be able to read the queries and understand what they do.
As far as “immediately” what you should look at is the badges “Trigger” setting.
There are various “immediate” triggers - eg. topic created - but the trigger for the First Share badge is “Update daily”. I’m not sure when “daily” happens, but I think it is midnight.