The very first forum I was a part of, they did something unique when they selected a new moderator. The new moderator was put into a separate group called "Moderators In-Training". When I went through the process, I could only give users earning and approve of posts. Over time, I gained access to other moderator tools. Before I was switched over to the main moderator group, I had to be vouched by other moderators.
I was surprised to see that other forums did not do this; instead, you were out into the main moderator group immediately. Stack Exchange has something similar to "Moderators In-Training", as users can unlock moderation tools as they gain reputation.
Since it seems uncommon to have a separate group for new moderators, how do other sites train their moderators, or do they train them at all?