I managed to successfully run to the capital city (Jeuno) yesterday without getting killed. (There's no such thing as corpse running in FFXI. You die and you either get rezzed by someone else or you respawn at your home point. Thankfully while you do lose some experience you get to keep all your gear unlikely earlier games.) My highest level is 20 which means I met the minimum criteria for doing the chocobo quests. As best I can tell there are 4 main things you can do with chocobos; 2 of which existed back when I played before and 2 of which are completely new to me and sound to be pretty awesome.
First of all, at level 20, you can start a quest in Jeuno to heal a sick chocobo. You need 4 pieces of Gausebit Grass and you have to visit a sick chocobo and try to feed him the grass 6 times. Each time has to be on a different game day, so it takes 6 hours of real time to do. Note you can do other things in the meantime and they don't have to be consecutive game days so it's not nearly as bad as it sounds. You just need to head to the chocobo stables 6 different times. Your reward for completing this quest is a license which lets you rent a chocobo from any of a multitude of places including all major towns and the main teleport locations. Renting a chocobo costs a small amount of gil (300-1200ish) and the ride lasts for 30 minutes or until you dismount. Chocobos are like mounts in World of Warcraft in that they are substantially faster than walking but are worse in that you can only get on at the designated locations and they cost money to use. They're better in that you can't get attacked while on one so you can freely run through high level aggressive monsters without dying.
Next up, while riding a chocobo you can feed it a gysahl green (vendor price ~70gil) to make it dig for buried treasure. There is a hidden skill level associated with chocobo digging (it's a lot like fishing except you get told your fishing skill in the status menu) which helps determine what you dig up. Also the zone in which you are digging and time of day can impact what you dig up. Also, every zone on has so much stuff in it so if other people are digging at the same time they can deplete the results. Finally, there is a fatigue system (again, like fishing) so there is a cap as to how often you can dig each day. 100 digs by default, 200 with some gear. Total cost to skill up to max is estimated at around 12.6M gil! The guide I found claims that once you've invested 300k you're pretty much guaranteed to make money as long as you're hitting the right zones and they aren't cleared out already by other people. I know I used to carry a stack or two of greens on me and dig as I was riding from place to place just for fun and to see what random junk I could find but it was at a loss.
They added a new mini-game where you get to actually raise your own chocobo. You can get a quest to get an egg at level 20 and then you get to play a life management simulator for the next _FOUR_ months. Choose what your chocobo is going to do each and every hour to manage stat gains and ability gains and whatnot. Think Princess Maker 2 but with fewer buxomize pills. At any rate, after about a month of raising a chocobo you get an item which lets you summon your chocobo to you anywhere in the world. So, you can now mount up wherever you want! This item has a limited number of charges and costs 400gil per charge to recharge it, so you don't get out of paying for your rides, but you can start from anywhere which is pretty sweet. On top of that the stats your chocobo has impacts how your mount behaves when you're riding it. You can make it go faster than the rentable ones, or ride for longer, or be better at digging. There's also a racing circuit where you can ride your chocobo for various rewards and the stats your chocobo has matters for there, too. You can even breed chocobos if you want, like in FFVII, and try to make different colours or better stats. You're only ever playing the life management sim with one chocobo at a time but it sounds like you can have multiple old chocobos lying around and switch between them as you want different stat mixes.
Finally they also added in the Chocobo Hot and Cold game from FFIX. You get sent to a specific location in the world and when you get there you can dig. You get told in relative terms how close you are to the treasure and have a number of tries to find it using just a vague starting location and the hints from your chocobo. Get it right in time and get a treasure chest with some random stuff in it. It sounds fun (I certainly fully completed the Hot and Cold game in FFIX both times I played it) but I can't tell from the potential item list if it's profitable or just fun.
In conclusion everyone should get a chocobo license for sure since that quest is pretty trivial and the rewards are huge. Beyond that having the ability to call your mount from anywhere seems really useful so hatching one personal chocobo and raising it seems good. If you don't pay attention to the life management game you'll still get to call your mount anywhere (well, there is a quest in the middle to get a whistle to do it so you'd have to pay enough attention to do that) but your chocobo would have low stats and be slower than a rental if you did that. But since you could still rent in towns it wouldn't be much of a hit. And if you're like me and actually enjoy life management games and random digging then chocobos seems like a pretty fun time sink. Must make a green chocobo...