To begin covering (breeding) racing chocobos, you must retire your first chocobo, which requires it being at least Rank 40. The guide begins at this point and assumes you know the basic mechanics of the system; see Chocobo Racing if not.
Once your first chocobo is Rank 40, visit the Race Chocobo Trainer at the Gold Saucer and choose Retire a Racing Chocobo to permanently remove it from racing status and receive a Retired Chocobo Registration G1-M or G1-F, depending on the gender of your chocobo. Don't lose this item, as it can be used to breed a retired chocobo a maximum of ten times.
Since you only have the one chocobo, you'll need to arrange a covering with one of the Gold Saucer's "breeding stock" by acquiring a Covering Permission from the Tack & Feed Trader to the left of the counter. Under the Race Items menu, you can purchase a registration for any grade (Pedigree Level) of either gender. The Covering Permission must be the opposite sex of your retired Chocobo Registration (you cannot use two Covering Permissions to breed a chocobo, at least one must belong to you).
A fledgling chocobo will always be one Pedigree Level higher than the lower leveled parent. You may cover your G1 with a G9, but it will still result in G2 offspring. (see: Choosing a Covering Permission) Once purchased, a Covering Permission will randomly generate a one-time registration for breeding. Every purchase will be different, and you can discard it if you don't like the stats. (More on that later.)
Accept the quest Like Sire Like Fledgling from the Chocobokeep next to the counter and teleport to Camp Bentbranch to complete it. Speak to Katering <Race Chocobo Creeder> again to arrange the covering. You can pick up your fledgling racing chocobo in six Earth hours.
What is passed on, and how? There is a bit of RNG involved, but with time and effort, it can be influenced and eventually reigned in. With hard work and good luck, you can breed the perfect chocobo champion (but even bad luck will get you close).
Known as the Pedigree (not the Pedigree Level), a fledgling chocobo will inherit its attributes' star ratings from its four grandparents. The star ratings of the two chocobos being covered are irrelevant. For each attribute, all four possibilities are thrown into the pool and one is randomly chosen. The goal is to slowly weed bad star ratings out of the pool entirely, which may require each new chocobo being covered several times, as fledglings with bad ratings can be released.
In addition to the ability each chocobo learns, birds that are bred also inherit a second ability from one of their two parents. This is as simple as a coin-toss. However, covering permissions with high pedigree levels have access to higher tier abilities; so long as you're willing to invest more MGP, you can attempt to introduce specific abilities into your chocobo's bloodlines.
Like the hereditary ability, a fledgling will inherit one of its two parents' feather colors by coin-toss. You might have to be pretty tenacious and burn a lot of MGP, but it's not impossible to tailor even the feather color of your champion racer.