After you finish " No. 1" mission, you get a call from Brucie. Since that moment, calling him and choosing the Race option will cause Brucie to tell you where the next available challenge can be found (follow the yellow radar blip).
The choice of the car is all up to you, but remember - the faster, the better. The race route appears on screen and on the radar.
You can always use the B button to focus your view on the nearby checkpoint. It is important not to kill your opponent. Otherwise, the race will be canceled.