After reaching a certain point in the main story quest "Descent into Darkness," a familiar black chocobo will arrive to Lestallum. Find him to the left of the power plant entrance to start his questline. After interacting with him, go and power up the new supply route to unlock Cape Caem. This will start the simple "Where Light Won't Reach" quest. After completing this quest, return to Lestallum and power up the new supply routes. Now you can return to Cape Caem freely, and it will act as a new hub where you have access to most of the stuff from Lestallum, in addition to a new chocobo farm.
Start by going to the stables via the elevator and do the training tutorial for your black chocobo. Select the stats you wish to train. This chocobo starts at level 50 and has a default cap of level 60. Most chocobos can only level up 10 times. To increase its level cap further, you will see yellow stat meters that fill after each session in the bottom right. If you fill out all these meters before you level up 10 times, you will increase your level cap. Sometimes there won't be a meter to fill out, thus the bird will automatically capout. If planned out right, with a good bird and a bit of luck, you should be able to get your bird to level 99.
Each time you select a stat to train, the chocobo will level up but careful so that it doesn't get exhausted, otherwise you will lose a levels worth of stat increases for each 20% of the exhaustion it has. You can keep an eye on the exhaustion % for each stat when selecting it. When the percentage turns brown, it means you are more likely to injure your chocobo if you try training that stat again without resting. You can let it rest at the cost of one level per 20% exhaustion.
Another thing that greatly helps is pressing the Y button to view your bird's motivational characteristics. The better the description, the better the stats of your bird. PRIORITIZE THIS.
when selecting birds to train as it will help getting the best stats possible. This makes filling out those meters less luck dependent.
A GAMER X360 took the time to figure what each stat determines below:
First sentence (max level):
"This bird roosts at the top of the pecking order!" - max level 99
"This bird's plumage exudes and air of prestige" - max level 80
"This bird's feathers have a certain sheen about them" - max level 50
"This bird is middle-of-the-flock in all regards" - max level 30
"This bird does not stand out from the flock" - max level 20
Second sentence (training affinity):
"It appears to possess incredible attributes." - bonuses for all, less injuries
"Its quick reflexes should help it sprint to and fro." - bonuses for speed
"Its powerful wings should allow it to stay aloft longer." - bonuses for jump
"Its sturdy musculature should make it more resilient". - bonuses for stamina
"It seems to possess no notable attributes." - rare bonuses, freq injuries
"Its sturdy musculature should make it more resilient." - better for Chocomostro? untested.
Also, there will never be an injury on the last training session of each level group, so you can take the huge reward with no risk.
Once its reaches its level cap, you will have to find a new chocobo by doing quests that have a chocobo symbol next to the recommended level. You can raise up to 10 chocobos before you have to start releasing the weaker ones into the wild. It appears to be random which quest gives you a chocobo. The level of the chocobo also appears to be random and unrelated to the quest. If you get a high level quest you can't beat, you start the quest and immediately quit and the chocobo quest should change.
EDIT: McAwesomePT provided an a great tip if you manage to get a bird with a good description and level:
if you blew up a good chocobo, DO NOT BACK OUT. quit the game without pressing B and you will have infinite chances to try again, as the progress will not save
Your chocobo will also be critical in unlocking all the royal sigils as it will need high stats to help unlock new routes. To help with the training, here are some of the dialog boxes that will appear after leveling up your bird:
"Your chocobo can't wait to get started! It will benefit greatly from its next session."
If you repeat the last session, it will give you extra stat increases. Make sure you pay attention to the exhaustion % here.
"Your chocobo appears eager to keep training. Your chocobo's exhaustion has decreased."
Greatly decreases exhaustion from all stats.
"Your bird seems pleased with that session."
Decreases exhaustion slightly for all stats
"Your bird seems satisfied with that session."
Decreases exhaustion from the stat trained??
"You notice a glint in your bird's eye."
If repeat the last training as it will give you great stats.
"Your chocobo is itching to spread its wings! You can now select the Motivation training course."
Same as the glint in your birds eye dialog. This will increase your stats for the next level up, but costs an additional level to motivate your bird.
"Your chocobo seems unenthused.. It will not benefit much from its next session."
Grants less stats gain if you repeat the same training.