Instructions
Travel to the Gold Saucer after ensuring that the Chocobo you want to improve is in its stable at the Chocobo Farm.
Enter the Chocobo arena in the Gold Saucer and talk to Esther, the woman at the counter. Choose to enroll your Chocobo in the next race.
Press "Select" during the race to switch to Manual control. Hold the "R1" and "R2" buttons to keep the stamina meter up, and hold "Square" to dash. You can also choose to have Esther explain the racing controls.
Win three races to advance your Chocobo's class to the next class, B-class. This applies to changing to A-class and S-class as well.