The first thing is that it is never too early to start. All of the drivers you see on TV every week, no matter the type of cars, started out young (some as early as 4 years old) at their local race track or in karts. The hard part is to prove that you have some ability there, prove yourself there and you'll quickly find yourself moving up through the ranks. Keep it up and you'll find yourself catching the eye of a big name car owner.
The First Step
Go to your local race track (dirt or asphalt doesn't matter) and buy a pit pass if possible. Then go in and strike up a conversation with someone. Drivers, crew members and officials are all great resources with different perspectives on what it takes to get started at that track. As long as they don't have pressing work to do most people will be more than happy to talk to you, but please be courteous.
Ask if they have a minimum age. Many tracks' age limit is lower than the state driving age. If your child is too young to race at that track then someone will probably direct you to a local karting association.
There are definitely no "gimmies" here. Hard work, practice, natural skill, luck and money all play a roll in your ability to catch a break.