I found the first race to be fairly easy to win but apparently you can not have the achievement pop if you fail the first time. To ensure you win the first race try this:
Start by hitting
or w/e the corresponding button is if you are using a keyboard (I have a controller). Then mash
(or corresponding buttons explained by Ester) so you are going max speed. About 2/3rds of the way through the race press
once to slow down but get ready to hold
right at the end. You should finish, with energy to spare and ahead of the pack.
If you fail then you get another shot and the race gets easier. Same strategy can be applied. If you win and it doesn't pop then don't worry. This can be earned later after you buy some stables from the Chocobo farm and catch your first Chocobo. Can be made even easier if you wait until you have a Golden Chocobo which can be obtained after collecting the items dropped by the Ultimate weapons.