1 Buy some greens
Greens enable you to keep the Chocobo busy while killing the enemies surrounding it. Make sure you buy some greens before encountering a Chocobo.
2 Equip Chocobo Lure Materia
The Chocobo Lure Materia increases the chance that you encounter a chocobo. Equip the Materia to anybody before searching for the chocobo. The higher the level of the Materia the better. See how to level up your Materia fast.
3 Walk over Chocobo tracks
Chocobo tracks can be found in certain areas in the world map such as the area near the Chocobo farm you encounter after leaving Midgar. Keep walking over those tracks until you encounter a chocobo in a battle.
4 Give the greens to the Chocobo
As soon as you encounter the Chocobo in a battle you want to keep it busy not to run away. Give the Chocobo the greens as soon as you get into the battle. See also how to learn new limit breaks fast.
5 Kill the enemies
In order to catch the Chocbo you need to kill all the enemies that accompany it. Keep attacking the enemies while the Chocobo is busy eating the greens you gave it.
6 Give the Chocobo more greens
If the Chocobo finishes eating it might get bored and leave. Give it more greens as soon as it's done eating. Note that more expensive greens last longer.
7 Never attack the Chocobo
If you attack the Chocobo it will get scared and leave then you will have to repeat the process all over again.
8 Avoid attacking all enemies
If you use a magic that is paired with the all Materia then the Chocobo will get attacked as well and it will leave the battle. See why the Materia system is great.
9 Buy the Chocobo lure materia
You can buy the Chocobo Lure Materia from the Chocobo farm in the first continent.