This article is from the Final-Fantasy and other Square Soft Games FAQ, by email@example.com (Nick Zitzmann) with numerous contributions by others.
To capture a Chocobo, you must have bought a 'Chocobo Lure' materia from
Choco Billy in the barn. Equip it to any character. Now, go outside and
wander around on the Chocobo tracks. Eventually, you'll enter a battle,
and if you're lucky, the music will be different and one of your enemies
will be a Chocobo.
To safely capture the Chocobo, kill off the other enemies, but don't
hurt the Chocobo. Work quickly, because if the other enemies start
attacking the Chocobo, it will go berserk after a while, attack
everyone, and then escape. You can prevent a Chocobo from running away
by using Greens on it (which you can also buy at the Chocobo Barn; they
come highly recommended). After killing all the enemies, the battle will
end and you'll find yourself riding a Chocobo!
As for crossing the marsh, since chocobos are much faster, it is easier
to avoid the marsh snake (Midgar Zolom). However, if the snake's shadow
touches you, you will still have to fight it. What's more, if you run
away, you'll find yourself back on the side you approached the marsh
from. You can get across the marsh without a Chocobo, but it requires a
lot of luck.