See first question in /Jeuno FAQ. The quest takes 6 hours.
6 hours?! Is it worth it?
Actually, I would say this is probably the most useful quest in the game (even more so than your subjob one at early levels). It lets you go places much quicker and with less risk of attack than on foot. You can use your Chocobo to dig for items. And it's relatively cheap. So I would say it's definately worth it.
Admittedly some quests in this game aren't really worth the effort. Some have low payoffs for a lot of work, and some take so long and show such little benefit in the short-term (subjob quest, I'm looking at you!) that you often only do them for completeness or fame. But the chocobo license is so freakin' usefull, especially if you've already spent the time walking to Jeuno, it's not even funny.
What is "Digging", how do I do it, and what do I get?
Digging is when you have your chocobo stop moving and peck at the dirt for stuff. You need Gysahl Greens for each time you want the chocobo to dig. You get various crap that you typically find in the area from fighting monsters.
Honestly, I'm yet to have this be worth the effort. I'm sure in some areas doing it will turn up rare items, but mostly I just get garbage.