I don't quite understand what you're trying to do. It seems to me that if the child can read your instructions, he/she is much too old to play with this kind of toy!
Do you always have to refer to the toy car as such? Can it be just car? Better to refer to toy. The same mechanism might be used in a toy truck, boat, tractor, train, etc.
Is wheels winding up okay? Again, not sure the purpose of this, so hard to give best advice. Maybe 'motor winding up' would be better.
Do you have a different word to describe the action other than go? Pretty hard to suggest something better than, 'It will go.'
Can I say push the car down too when you're not really pushing the car down but putting just a little bit for downward pressure to make sure the wheels crank? You could do that, but I suspect a child will quickly discover the need for a bit of downward pressure to prevent wheel slippage. No need to over-think things.
If you are writing instructions to go on a toy package for parents to read, I suggest something like:
'Place the toy on a smooth, level surface. With enough downward pressure to prevent wheel slippage, pull the toy back until a clicking sound is heard. That means the spring is fully wound. Release the toy and it will go!'
Student or Learner