The starting NPC will be at the Chocobo stables and will ask you to deliver a Chocobo to a distant stable.
If you accept the assignment, you will be placed on a Chocobo and will need to ride to the destination.
You cannot dismount the Chocobo and still complete the quest.
Your reward will depend on how quickly you deliver the Chocobo.
Note: It is extremely difficult to meet the time requirement to be awarded a Miratete's Memoirs. Several other "Chocobo Riding Game" routes have far more lenient requirements to get an XP scroll.
You will not be offered this quest if you have attempted any of the other Chocobo riding quests this week. This includes the San d'Oria, Bastok, and Windurst quests.
This quest is not always available. MithraPride has a schedule of when the various origin/destination combinations are available, and although the timer does not specifically mention this route, this route is offered on the same schedule as the "Bastok Mines -> Windurst Woods" route.