I would say that it is up to the parent's judgement rather than any law. In our city, there is a summer safety town for kids that are kindergarten age (about 5 years old). They learn about bike safety and well as pedestrian safety, etc. Other than that, there is no bike training that I have heard of.

My DD is 11 and she can ride her bike to her friend's houses in the neighborhood. There are 525 homes in our neighborhood so it is a fairly large area with a park in the middle. I do not let her out of the neighborhood on the busier streets alone.

Usually, going on the busier streets to go to the stores or restaurants occurs around age 13 here. There are bike lanes paved on the side of the roads, but there is no grassy area separtating the road from the bike path for the most part so you have to be a competent biker who can stay in the small lane and not get freaked out by fast moving traffic nearby.