Once you have gained level 20 you can obtain this quest. To start the quest you must travel to Jeuno, which is no easy feat on your feet.
From Bastok: Bastok Markets/Bastok Mines > South Gustaberg > North Gustaberg > Konschtat Highlands > Pashhow Marshlands > Rolanberry Fields > Lower Jeuno > Upper Jeuno
From San'dOria: South or North San'dOria > West Ronfaure > La Theine Plateau > Jugner Forest > Batallia Downs > Upper Jeuno
From Windurst: Windurst Woods > East Sarutabaruta > Tahrongi Canyon > Meriphataud Mountains > Sauromugue Champaign > Port Jeuno.
Most of us by level 20 have been as far as Konschtat Highlands, La Theine Plateau, and Tahrongi Canyon. The zones past those are full of monsters that will eat level 20 characters so be very careful and avoid having them see, smell, and hear you.
In Upper Jeuno you talk to Brutus and he mentions getting a chocobo to like you before he will give you a license. He also asks a couple questions of you. You will see a chocobo (well you will see 4 :P) with a boy next to him by the name of Oskar. Talk to Oskar and he will tell how the last owner left this poor chocobo and he needs Gausabit Greens to help him, though they aren't very tasty (what medicine is?) so you will have to earn the chocobo's trust.
Now you are probably wondering about these Gausabit Greens. They are found in Dangruf Wadi off Wadi Hares. Dangruf Wadi is off South Gustaberg, near Bastok. You will need 4.
So you have your Greens, you trade one per Vana'diel day (one hour real time) to the chocobo for 6 days. (each time the Vana'diel time is 00:00 you can do another turn in) You only need 4 greens because at first he doesn't trust you or your nasty greens. Doesn't need to be consecutive.
After you have completed this Brutus will find you to be good to chocobos and you obtain the right to rent chocobos. Congrats!
Additionally, this is also the beginning of the Advanced Job Quest for the Beastmaster class.