FAR [§ 61.83][1] states: 

> To be eligible for a student pilot certificate, an applicant must: 

>(a) Be at least 16 years of age for other than the operation of a glider or balloon. 

>(b) Be at least 14 years of age for the operation of a glider or balloon. 

>(c) Be able to read, speak, write, and understand the English language. [...]

A **student pilot certifacte** enables you to fly a plane solo or on a solo cross country.
In other words, you need to be at least 16 years old to fly a plane (non glider/baloon) solo. 

![This is an example of a student pilot certificate. It comes with your medical usually.][2]

Now, back to your initial question. If you go through FAR [§61.83 - §61.95][3] you will find all the regulations about student pilots (which would be the kind of pilot you are referring to ). 

I have found **no age restriction** going through them for the pre-solo instruction. All it says that it must be done before soloing which does require a medical ([but these don't have age restrictions either][4]).

**Summing up, a 9 or 12 year old child is legally able to take flying lessons with an instructor, but may not solo before his 16th birthday (no glider/balloons).**


  [1]: http://www.flightsimaviation.com/data/FARS/part_61-83.html
  [2]: https://i.sstatic.net/SlCD6.jpg
  [3]: http://www.flightsimaviation.com/data/FARS/part_61.html
  [4]: https://www.faa.gov/licenses_certificates/medical_certification/faq/response7/