Chocobo Farm (Disc 1) - FF7 Walkthrough
At the Chocobo Farm you'll want to speak with Choco Bill who is in the northwestern house. He'll tell you to go speak to his kid in the barn who is named Choco Billy. This kid will give you the run down on just about everything you need to know about Chocobos as well as sell you the Materia you need for catching them, the Chocobo Lure Materia.
You'll also need to purchase some greens from Billy too - these will keep the Chocobo occupied during battle long enough for you to defeat the other enemies. Usually the more money you pay for greens the longer it'll keep the Chocobo occupied during combat. Here's a quick run down of each green and what they do...
Mimett Greens: Chocobo eats it for 1 turn and also allows the Chocobo to use Chocobuckle.
Gyshal Greens: Chocobo eats it for 2 turns
Tantal Greens: Chocobo eats it for 3 turns
Curiel Greens: Chocobo eats it for 5 turns
Krakka Greens: Reduces the Chocobo's chance of running to 33.33% for each battle
Pahsana Greens: Reduces the Chocobo's chance of running to 20% for each battle
I highly recommend purchasing 5 or more of the greens that you plan to use. There's a good chance you will end up wasting one or two because a Chocobo either runs from battle before you can throw it at him or you just get unlucky and take too long defeating the other enemies during the battle.
Before you're done at the Chocobo Farm you'll want to interact with one of the Chocobos in the pen and tell it 'Wark'. This will net you the Choco/Mog Materia which is a pretty useful summon for this early in the game. Once you've acquired that you're done all you can do at Chocobo Farm, exit to the world map again.
Now what you're going to want to do is venture over to the nearby Chocobo tracks and equip the Chocobo Lure Materia that we purchased earlier. Run around ontop of the Chocobo tracks until you get into a random encounter, if you were ontop of the tracks when it happened a Chocobo will be in the battle with you.
Go into 'Items' and select whatever greens you purchased earlier and use them on the Chocobo. Do this very quickly or the Chocobo will run from battle on you. When you give a Chocobo these greens he'll stop what he is doing and eat them - he also won't run from battle until he is done eating.... Assuming you don't hit him of course. If you hit him he leaves - so no AoE attacks!
Note: It's recommended that before you leave this area you pick up Lv 4 Suicide Enemy Skill. You can get it from a Mu, they're the little gopher looking enemies. Your character will have to have a level that is a multiple of 4 to have Lv 4 Suicide cast on them.
If you defeat all other enemies in battle while the Chocobo is still eating the greens you'll successfully catch him and you will be riding him when you return to the world map.
There are ways of crossing the swamp area that the Midgar Zolom patrols other than getting a Chocobo. One of the most common methods is to enter the menu every time the Midgar Zolom gets close to you and save the game then reload your save file. He'll be somewhere else in the swamp and you'll have a few seconds to run towards the Mythril Mine again before he gets close. Repeat this two or three times and you'll make it to the mine.
As for the second method... well, if you time things just perfectly you can just run across the swamp and totally avoid the snake! Here is a video of me dodging the Midgar Zolom without a Chocobo.