*This solution may contain spoilers*
After the events of Bevelle and watching the under water date scene you'll arrive and have access to the Calm Lands! A nice massive open grass plains, if you work you way towards the middle of the map you'll find a save point and a shop. Move north west from here and work you way towards a lady sitting on a chocobo looking outwards into the horizon. After talking to her about chocobo's ask her to train one, you'll then be given 4 courses to complete.
Wobbly chocobo
Dodger chocobo
Hyper dodger chocobo
Catcher chocobo
These time trial courses involve beating a pass time before being able to move onto the next course. With some practice and a little bit of luck you'll have no problem passing them! After completing the Hyper dodger chocobo course pass time you'll unlock Catcher chocobo, which is a race against the trainer, if you manage to win the race, trophy unlocked, job done!
Just a small note if you contuine on with the story and obtaining the airship, the trainer will have moved right next to the shop in the middle of the calm lands. About 10 steps south from when you've teleported to the calm lands.