I am in this exact position (graduating college in 8 weeks), so I guess I'll give this answer a shot.
I think that you need to encourage you child to find activities where he/she can learn how to learn by himself/herself. For me, that was debate. We had no coach, we had no teachers, it was just a bunch of kids and books, taking on the best teams in the nation. Or perhaps the Robotics club, or some sort of community service outlet, these are all great activities that build great people and self-starting skills.
You can also encourage him/her to figure out much earlier what his/her passions are, if he's/she's young enough, perhaps get him/her to read books on different professions and encourage him/her to learn about them by himself/herself.
As the internet becomes more pervasive, it has become a lot easier to learn online and off, but at the end of the day, I think that being a self-starter requires a lot of support from friends and family and the drive and passion to get what is important. So even by asking this board, you children should be in good hands.
PS. I went to public school, and it was an awesome experience.