For a quick start, use Quickman's design (the V-like item in his car.) Hold X after the first light.

Keep racing against your opponents until a sign appears saying "HERE COMES A NEW CHALLENGER." Beat the challenger to receive one part of the secret car. Continue to do this until you've collected all four parts.

Here's where to find them:

DESIGN  - NAPALMAN'S STAGE
WHEELS  - QUICKMAN'S STAGE
BOOSTER - ICEMAN'S STAGE
BODY    - GUTMAN'S STAGE

NOTE: The locations above work for all characters except for Gutman, Quickman, Iceman, and Napalman.