Acquiring a Carob Nut
You have the birds, you now just need to give them the proper nut. Unfortunately, you can’t just buy the nuts you need, you’ll have to snatch them from monsters! Fly to some grassy islands between the western continent and the northern continent (north-west of Costa del Sol) where you can find the Vlakorados foe (big red dinosaurs). They’ve got a ton of Hit Points, but otherwise aren’t too much of a threat; just steal from them until you get a Carob Nut. You’ll want to repeat this process at least three times… but there’s a good chance you’ll need more, depending on your luck (or lack thereof!)
Breeding Green and Blue Chocobos
Time to make some babies! With four Good or Great Chocobos (two of each gender), several Carob Nuts in hand and a massive stock of Sylkis Greens (at least twenty for each Chocobo), it’s time to head back to the Chocobo Ranch. That is, if you don’t want to race Chocobos yet. It’s not required to get the first two breeds of Chocobo - Green and Blue - but it will increase your odds. If you DO want to race your Chocobos, check out the "Race For Rank!" section, below.
Note: Save your game before feeding Chocobos, as feeding them different amounts of greens will result in different stats, and hence, different outcomes when breeding. It’s essentially a way to quickly save/load your breeding attempts and speed up a luck process.
Anyways, feed every Chocobo twenty Sylkis Greens then pick a male and female you want to breed; again, either two Great Chocobos, two Good Chocobos, or a Great and a Good. Once you’ve fed them greens and selected them, confirm you want to breed them and give them a Carob Nut. Shortly you should have a brand new Chocobo, hopefully either a Green Chocobo or a Blue Chocobo. If you get another mundane one, you have two options; reload the save you made before feeding the Chocobos and feed them different greens (twenty-one to one of them) or breed different Chocobos, which will hopefully re-roll your results, or failing that, fight around a dozen battles and try breeding the birds again.
When you get a Green or Blue Chocobo, you can breed that colored bird with another Good or Great Chocobo (even its parents! Yuck!) and hopefully get a Chocobo of the other color… or you can try breeding its parents, or another set of Good or Great Chocobos, it’s all rather luck whether you get a Blue or Green Chocobo, so keep at it! Personally, luck seems to favor getting a Green Chocobo first, then a Blue Chocobo from breeding the Green Chocobo with a Good or Great Chocobo.
Green Chocobos can cross mountains, allowing you to access Lucrecia’s Cave, the Mime Materia cave along the southern end of Wutai island, and the Ancient Forest (see the included map, above). The Blue Chocobo can be used to cross the shallows around the archipelago north-east of Mideel, where the Quadra Magic cave is located.