Ok, here I go I'm going to tell you the whole process of breeding a golden chocobo.  Here it goes.  First of all you should know that it's going to take a lot of money and time to get a golden chocobo.  Expect to spend somewhere around 3 hours and a couple hundred thousand gil.

    First you need to purchase all 6 of the stalls that they are offering at the chocobo farm.  They will cost 10,000 gil each so it'll add up to 60,000 gil.

    Next you'll have to know that you can catch chocobos by walking over their tracks with the chocobo lure materia equipped you can either buy this materia from chocobo Billy or find it by the outside right portion of the outside chocobo pen.  Now there are several locations where you can find chocobo tracks, however only a few of them are of any use.  To see where the tracks of any use are check out my  MAP .  To capture the chocobos you will enter a battle with the normal enemies of that area plus there will be a chocobo.  DON'T HIT THE CHOCOBO!  If you hit the chocobo it will attack everybody and run away.  It helps drastically to feed the chocobos greens to keep them occupied while you kill the other enemies.  You can buy these greens either from chocobo billy or the chocobo sage himself.  They vary in price but the more expensive the better they are so you're best bet is to buy the expensive ones right off the bat.  You can get away with only buying one green and duplicating it with the W-Item Materia by using the duplicate item trick to stock up and get 99 of the green of your choice.  If you do get the chocobo (which isn't all that hard to do) you will be riding it when the battle ends to dismount press X or push the direction buttons to ride it.  If you have a stall when you dismount it will ask you where to go it will say let it go or send it back to the stables.  You want to send it back to the stables (if you don't how are you going to breed it?).  You can only hold up to 4 chocobos outside before you have to go back to the stables and tell em' which ones you want to put in the stables.  You only want to put in the best chocobos because when you breed the little ones inherit what their parents can do.  Here is how to tell if the chocobo is good or not.

*useless* Standing chocobos (they will be standing and swaying), also chocobos that are walking with head down are useless
*walking* Head up walking
*running* Head up running
*dashing* Head down sprinting very quickly

    I suggest that if they are standing still or not moving get rid of them because they are worthless.  When you finally get the chocobo of your choice then say that you want to put it in a stall and he will tell you what the gender is.  We are breeding chocobos so gender is VERY VERY IMPORTANT.  You can name them whatever you want to so have fun with this one.  Just make sure you know where they are from because when you put them in the stall you can't tell what kind they are like DASH2 for your 2nd dashing chocobo.  Here are the good types of chocobos and where they are found

Walking chocobos:Southeast of Gold Saucer
Running Chocobos:Southern Island where Cloud was ill
Dashing Chocobos:Western Side of the Northern Continent
 

Not all of the chocobos found in these areas will be good for breeding so make sure they are good ones before you breed them.

Here is some info on the chocobo sensei.  He is found on the northern continent, the one shaped like a chocobo.  The chocobo sensei is a man who will sell you the best greens in the game for you to feed to your chocobos.  If you talk to his green chocobo that is inside his house you will receive the yellow Enemy Special Materia.  His house is near the center of the continent.  It looks like a regular house with snow all around except close to the house there is grass so you can land the airship.

To race your chocobos go to the gold saucer in the chocobo square and talk to Ester, the girl in the left corner, and she will let you race.  You should be able to figure out the rest from there on.  If you win you can either get GP or an item.  The items get better when you win more races and your chocobo gets in a better class.  There are 4 classes they go like this:  C-B-A-S.  You have to win 3 races in a class to move up to the next class.  If you win something like 5 or 8 races in S class Ester will say that you are the best racer she has ever seen.  She will give you some very nice items for being such a good chocobo racer.  If you hold down R1&R2 your energy meter will not go down as slowly so you will win much more races more easily.  If you do this you won't have to feed your chocobo as many greens to get them better.

Now that I've covered the basics here is how to start the breeding.  First you will have to breed the green chocobo and the blue chocobo.  The green chocobos can run over mountains and the blue chocobo can run over river or shallow water.

First capture 4 chocobos a male and female walking and a male and female running.  make sure they are the right kind because it's crucial.

Now that you have the first 4 chocobos you need to get the right nut to breed them with.  To get it go to the chocobos Sensei's house and equip the yellow steal materia.  Run around on the grass and only the grass until you encounter a large red dragon.  You'll know when you find one because the only other enemies are little squirrels.  When you get in the battle with the red dragon start stealing from him eventually you will steal the nut you need to breed them with.  If you think you're done now you're wrong you need to steal 3 nuts from the red dragons by the chocobo sensei's house.

Once you have the nuts go back to the chocobo farm and save your game before you go in.  Then go the chocobo billy and select breed chocobos.  Select your male walking and your female running.  Next he will ask you which nut you're going to use select the one you just stole from the dragons.  It should be the second from the bottom and have a 3 by it indicating how many you have.  If you succeed you will have a green chocobo that can cross mountains.  If you don't succeed reset your game and load the one you saved right before you entered the chocobo farm, and repeat the breeding process.  When you get the green chocobo go outside and save your game.  Then go back inside and choose the other two chocobos female walking and male running.  Breed those two with the same nut and you should get a blue chocobo that can cross rivers and shallow water.  If you don't get the blue one then reset the game and load the one that you saved before entering the chocobo farm the second time.  When you get one of each color make sure that they are of different genders.  If they aren't keep resetting and loading the game until you get the desired color and gender of both.

Once you get the blue and green chocobos of different genders you can't breed them for a while so now is a good time to go explore with your new chocobos.  You may also want to go up north in the snow and capture 4 chocobos because you're going to need them in a while.

After a while go back to the chocobo farm and save your game.  Then enter the farm and you should have one of the nuts left that you stole from the dragons in the beginning.  Now breed you blue and green chocobos with each other with the same nut as before.  Eventually you will get a black chocobo that can cross both mountains and shallow water.

Next is the golden chocobo.  It takes a little more work to get him but when you do you can finally get the knights of round summon materia.  First look at the 4 chocobos you captured earlier and get one that is of opposite gender of your black chocobo.  Then you need to race your black and yellow dashing chocobos until they are both in the S class at the gold saucer.  When you have them in the S class you need to ge another special nut.  It is located in the northwest corner of the map.  It is on the map (not the one that isn't on the map).  It has some forests and a place where you can land.  Run around in the forests with your yellow steal materia equipped.  You should encounter some little goblins with boxing gloves on.  You can get the nut you need by stealing ti from them.  You only need one of these.  The go back to the chocobo farm and save your game before entering just like before.  Go to chocobo billy and select breed chocobos and use the nut that you just stole it should be the last nut to choose from.  Breed your black chocobo with your yellow dashing chocobo that is in S class and you should get a GOLDEN CHOCOBO.  Some people say how can you tell the difference between a yellow chocobo and a golden chocobo.  It's easy the golden one is very very golden.  If you don't get it on the first try the reset and breed again.

Once you have your golden chocobo you can get the knights of round summon materia in the northeastern corner of the map.  It's in a cave on the island that isn't on the map with nowhere to land so you have to walk all the way there on your golden chocobo.  Also with your golden chocobo you should be able to win an S class race and you will get a very nice selection of items.  Or if you prefer you can get the GP and trust me the items are worth quite a bit of GP.