Many professional race car drivers started by driving go-karts at an early age. Go-karts are one of the most accessible types of race car there is as many towns and cities have their own tracks.
Keep going back and improving your driving. When you reach a high enough standard you may be asked to join in local go-kart races. The key of developing through the racing ranks quickly is getting noticed.
To be a professional race car driver you need to be in great shape as it is a very physically demanding job.
Read and watch as much about racing as possible and apply what you learn to your racing.
To be a good racer you have to know a lot about cars so never miss an opportunity to get involved. Ask your parents if you can help them maintain their car. Like checking the tyres, oil, water and so forth.
Get yourself down to your nearest race course and start talking to the course staff and see if you can get access to the race teams. If you keep hanging around and asking questions they may let you help out as part of the team. You won't get paid for this and you will be doing the mundane jobs but it is where most people have to start.
Look for work experience in car related companies like garages and repair shops this will again increase your knowledge of cars.
As soon as you are old enough to legally drive on the road start getting lessons and get your licence. The public road is not a race track so don't treat it as one but the things you learn from a road car can be brought across to a race situation.
Save up money from doing extra jobs and attend driving schools and driving days. Try and get as much expert tuition as possible.
Try your hand at as many different styles of cars and racing until you find what suits you the best.
You may want to start your own small race team to enter in to local events. This is where it comes in handy to have friends that are in to cars and a company that will sponsor you.