Anyone who wants to breed Chocobos needs to find a Male and Female Chocobo Card to breed as well as obtain a VCS Honeymoon Package.

Keep in mind a couple of different factors when collecting your 2 Chocobo Cards:
-  2 Cards of the same color greatly increase the odds that your Chick will become this color.  Kind of logical.
-  If you want your Chick to be born with an ability, at least one of the parent cards must bear the desired ability.  Chances greatly increase when both parent cards bear this same ability.  Kind of logical as well.

Your next step is to visit Finbarr at the Upper
Jeuno Stables.  Finbarr is the NPC which sells Honeymoon plans for the thrifty price of 5k per plan.  Each honeymoon plan has a certain function which may either be increases the chances that your Chocobo is a certain sex or adding a slight advantage to a base stat.

Listed here are the 4 types of Honeymoon packages that you can choose from.

A. Gourmet Plan
Base Stat:  Strength
Increased Chance of Gender:  Male
...the chocobos can enjoy fine dining together.
Good food gives the chocobos strength, and increases the chances that a male chick will be born... increased chance the resulting chick
will become male, and inherit abilities from the parents.

B. Sports Plan
Base Stat:  Endurance
Traits inherited from:  Male Parent Card
...the chocobos can enjoy a vigorous exercise program together. This increases the chance
that a chick with high endurance will be born. Competition also raises the possibility that the male chocobo's traits will be inherited...
increased chance the resulting chick will become male, and inherit color from the parents.

C. Hiking Plan
Base Stat:  Discernment
Increased Chance of Gender:  Female
...the chocobos can enjoy the sights and sounds of nature together. The fresh air will have a beneficial effect on your chocobo's intelligence, and increase the chance that a female chick will be born... increased chance the resulting chick will become female, and inherit abilities from the parents.

D. Jeuno Tour Plan
Base Stat:  Receptivity
Traits inherited from:  Female Parent Card
...the chocobos will get a tour of Jeunoan
streets led by yours truly. Seeing new sights will improve the chances of a highly receptive chick being born. Another of this date's features is the increased possibility that the female's traits will be inherited... increased chance the resulting chick will become female, and inherit color from the parents.


Trading a VCS Honeymoon Package Card plus your two Parent cards results in an adorable Cut Scene where the two Chocobos corresponding to the two Chocobo Cards you traded appear and go on a honeymoon together.
It's probably one of the most adorable Cut Scenes in the game, so Enjoy!

After the Cut Scene, Finbarr says a standard line about how he wonders what color a Sire
and Dame's offspring would be.  This is just a standard NPC quote and has nothing to do with your egg.
 

Wait approximately a Real Life 24-hours and return to Finbarr and he should grant you your new Chocobo Egg.  I've seen the egg being granted a few hours less, so this is just an approximation.


Now it's time to bring your egg to a VCS Caretaker in one of the home villages.
Note:  Whoever you choose to be your Chocobo Caretaker will remain this Chocobo's caretaker for the remainder of its existance in that particular Stables.
You cannot check another Stable to feed and take care of that Chocobo.  This village will be your permanent location where you will raise your Chocobo.
After this Chocobo is retired and you begin a new Chocobo, you may designate another village's stables as the location to raise your Chocobo.  But once again, you cannot change Stable locations midway during the Raising Process.