Our favorite ostrich-lookalike, the Chocobo, has been a part of the Final Fantasy mythos since FF2 (FF4 Japan)...  FF2 introduced the yellow chocobo
as a creature you could ride around on.  Not especially bright, but fast
and cute with annoying theme music.  There was also a white chocobo that 
could heal your MP when you talked to it, and the Big Chocobo (Lali!) that
would hold onto your items for you.  A special Black chocobo could even fly
over oceans and land in forests.  It was even smart enough to return back
to it's home by itself.  FF3 (FF5 Japan) introduced a 3D perspective when
riding the birds, but lost all the chocobos except the yellow one.  Now,
in Final Fantasy 7 chocobos reach their full potential!  Fans of the 
stupid birds unite!  FF7 allows you to catch, raise, race, ride and breed
chocobos with a variety of special traits.  Why bother?  Colored chocobos
that you have bred yourself can get to places your airship can't land.
There are potent magics hidden away in caves around the world....
 
Catching Yellow Chocobos:
In order to catch a Yellow chocobo, you must have the Chocobo
 lure materia equipped.  When you walk on the chocobo tracks
 there is a chance you will fight a battle with a chocobo and
 it's 'escort' of monsters.  If you kill all the monsters
 without hurting the chocobo, you'll catch it.  If you own a
 stable, you can send it back there when you get off (X button).
 If you don't own a stable, it just runs away.  Tamed chocobos
 (ones you have named) don't run away when you get off, they
 just sit there until you get back on.
Note:  If you take too long fighting the battle, the chocobo
 will run away.  To keep it occupied, feed it Greens in the
 middle of the battle.  This is one of the few uses for Mimett
 greens (since feeding them to stabled chocobos is pointless).
 Don't use Sylkis greens (or another expensive one) to catch
 the chocobo at 5000 gil a pop!  Mimett and lower work just
 as well.  Most of the time, you can expect to kill the
 escort before the chocobo has a chance to run away.  If
 you're worried (like in the Mideel area, the bastard great
 chocobos seem to have a 2 second attention span) you can
 try using Odin (he won't kill the chocobo, but the rest of
 the enemies are auto-killed.)
There are at least five types of yellow chocobos you can catch.
 All will start out at rank C, so don't worry about that.
 When you tell Chocobo Billy to move it to your stable he
 will make a comment about them.
Wonderful chocobo  ("This.. is a wonderful chocobo!") - 
  When moving it to your stable, it looks like it is
  dashing with it's head way down.  (Like S class chocobos
  look when you choose to ride them).
Great chocobo  ("This is a great chocobo.") - Looks like it's
  running when you move it to your stable.
Good chocobo  ("This is a good chocobo.") - Looks like it's
  walking.
Average chocobo  ("This is an average chocobo.") - When moving
  it to your stable, it is standing still.  Don't bother,
  just release it if you catch it.
Crappy chocobo  ("I really can't recommend this one.") - This
  one walks slowly with it's head drooping.  Turn this one
  loose.
And here are the places you can catch them and what "escorts"
 are with the chocobos you want.
Area next to the Chocobo ranch:
   All chocobos are poor (not recommended).  Don't bother catching
   chocobos here.
Area west of Fort Condor:
  Chocobos here range from poor to average.  This is also not 
  worth the effort it takes to catch them.
Area east of the Gold Saucer (near the peninsula):
  Several parties of chocobos contain good chocobos.  The rest are
  average chocobos.  Figure about 50/50 odds you're fighting a "good"
  chocobo.
Area west of Mideel (on the same island):
  The chocobo accompanied by ONE armadillo-creature is a great chocobo.
  The rest are average.
Area south of Icicle inn (snow):
  The chocobo accompanied by two bunny-rabbit/jumping enemies is a
  wonderful chocobo.  The rest are poor.
Riding chocobos:
To ride one of your tame chocobos, head to the chocobo farm and
  tell Billy you want to ride one.  Press X to get off, O to get
  on.  Once you get off a wild chocobo, you have to either send
  it to the stables (if you have one) or let it go.
Breeding chocobos.
You don't need much skill to breed chocobos, just a LOT
  of luck and patience.  NOTE: You can't start breeding until Disc 2 (when
  you can buy stables).  You can't race your chocobos (making it extremely
  hard to get the Gold chocobo) until after you get Cloud back in Disc 2
  after the first two huge materia quests.
Preparations:   You'll eventually want 40 sylkis greens, 3 carob
  nuts, 1 Zeio nut and 4 stables (minimum).
Sylkis greens: Visit the Chocobo sage's house.  Sylkis greens are
  5000 gil a pop, so 40 will cost you 200,000 gil.  You don't need
  them all right away (you'll need twenty at stage two and another
  twenty at stage three.  If you're cheap, you might get by without
  them at stage two (saving you from having to buy 20 = 100,000 gil),
  but you'll need all the help you can get at stage three...  If you
  have money to burn, get 60 (300,000) or 80 (400,000 gil).  It's not
  really necessary though.)
Carob nuts: After you get Cloud back, you can buy them at the
  gold saucer wonder square for 500 GP.  Or you can steal them
  from the red vrakoladus (a red dragon-looking enemy found in
  the green near the chocobo sage's house or near bone village).
  The vrakoladus has an obscene amount of HP, able to survive
  seven or eight Ultimas or Neo Bahamuts.  It's a better idea to
  run away after you steal the carob nut from it.  You'll need
  two carob nuts at stage one and another at stage two.  You
  could pace your time, get two and come back later while you're
  waiting for your chocobo chicks to mature.
Zeio nut: There are several islands in the NE corner of the world
  that your airship can land on.  One has a little patch of forest.
  Land on that and run around until you fight goblins.  You can
  steal Zeio nuts from them (it's hard though) or you can occasionally
  win them in battle.  You only need one (at stage three) so you could wait
  for later.  (Goblins are also on KOTR island, but if you're fighting
  them there, you don't need the Zeio nut!)
Stables:  Stop by the chocobo ranch and talk to the father (in the
  house on the far left).  Tell him to "take care of your chocobos"
  and buy 4 stalls.  If you buy five stalls, you can keep all the
  colored chocobos (as souveniers I guess), if you buy six you
  can even keep one of the chocobos you use in Step 1.  Why bother?
  Nostalgia I guess.  ;)  I'd save the 20,000 and just get 4.
Stage 1.
In stage one, you're going to mate a Good chocobo and a Great
  chocobo twice to get a Mountain chocobo and a River chocobo.
Head off to the Gold saucer, land and catch three chocobos and
  send them back.  Odds are one of them is a "good" chocobo.
  Now fly to Mideel, and wait for the "great" chocobo to come
  along.  Catch it (probably with Odin) and send it back to
  the stables.  Read the bit about which chocobos are escorted
  by which enemies to find the right ones.
Head back to the Chocobo farm and tell Billy you want to
  move your chocobos.  DON'T move them yet.  Just view them
  with the left/right directional buttons.  You should have
  at least one walking chocobo and one running chocobo.  If
  you have the standing-still average chocobos, let them go.
  If you don't have the walking chocobo, head back to the
  gold saucer and try again.  If you do...  cancel the
  move and save your game.  Now go back and move them.
  You want one each of the walking and running chocobos and
  you want them to be opposite genders.  Although the type
  of chocobo (good, average, great, etc..) is set when you
  catch it based on the escort the chocobo is with, the
  gender is not.  It isn't determined until you actually
  move it to the stable.  If you end up with the wrong
  genders, reset and try again.  Once you get the genders
  you want, save again.
If you bought 80 sylkis greens: Feed each bird 10 greens.
  If you bought 60: Either feed these birds 10 greens apiece
   and re-use them in the second part of stage 1, or wait
   until the second stage to feed those chocobos 10 apiece.
Now mate the two you caught, feeding them a carob nut.
  In about 20-25 tries (with unfed, un-raced chocobos) you'll
  get a Blue or Green chocobo.  Name your little baby warker
  and save the game.  If you get a yellow chocobo, reset
  and try again.
IMPROVING YOUR ODDS: If you want better odds of getting
  the blue or green you can race them.  Improving their class
  helps, but I'd figure the time it takes to race them could
  be just as easily spent resetting-and-retrying.
Once you have a colored chocobo, you'll need to do this
  whole thing again.  If you bought 60 greens and fed ten
  each to your chocobos, you might be better off waiting
  until they can mate again (this takes a while).
If you only have four or five stables, you either have
  to wait until your chocobos can breed again, or you have to
  let them go and catch two new ones like you did before.  If
  you have six, you can catch two new ones without releasing
  the old ones, but you'll eventually have to let three yellow
  chocobos from this stage go.
However you do it, repeat this stage again and get the
  chocobo color you didn't get before, of the opposite
  gender.  This takes a little longer obviously.  Viva coffee!
Stage 2.
You got a green chocobo and a blue chocobo and they're of
  opposite genders?  Bravo!  Now go kill some time because
  they can't mate right away.  If you only got two carob
  nuts before you can go get one now.  It would be REALLY
  nice to have 20 sylkis greens now, so if you don't have
  them, kill some hapless monsters and extract their money
  until you can afford them.  Of course you could do
  without them but this is much harder than getting a blue
  or green.  If you have 4 or 5 stables, you need to sack
  all the yellow chocobos at this point.  If you have six,
  you can keep one as a souvenier.  IMHO, I don't think
  they're worth it.
Wander around, splatter monsters and kill time until your
  chocobos can mate.  If you have at least 20 sylkis greens
  at this point, feed 10 each to your blue and green
  chocobos.
When you can, mate your green and blue chocobos with
  a carob nut.  With lots of luck, you'll get a black
  chocobo (gender doesn't matter).  You might also get
  more blue or green chocobos, or a yellow chocobo.  Reset
  if you don't get the black one.
I was trying for the gold chocobo before I got Cloud back,
  so I couldn't race them.  I still got the black chocobo
  in about 30 or so tries.  Racing them can really up the
  odds though...
Stage 3.
You have a green, blue and black chocobo?  Spiffy!  If
  you have 4 stables, you need to sack either the green
  or blue one.  If you have 5 or six, you can keep them
  both as souveniers!  They at least look cool.  I think
  the green one looks cooler than the gold one anyway.
Now you need to catch a "wonderful" chocobo.  You could
  have caught it earlier and left it outside the stables
  if you wanted to, but catching it gives you something to
  do while you wait for the black chocobo to mature.
  Once you catch it, you want one of the opposite gender
  of your black chocobo.  Do the save-and-reset trick to
  make sure you'll get it.
Read the bit above about escorts and finding the right
  chocobo.  Splatter more monsters, interrogate the chocobo
  sage some more, play the stupid Mog game at the gold
  saucer or do whatever until your black chocobo matures.
It's practically impossible to get the Gold chocobo
  from unfed, un-raced class C chocobos.  Even feeding
  them only helps a little.  I tried for 6 hours (over a
  period of a few days) to get the Gold chocobo before
  I got Cloud back and never got it (several hundred
  attempts.  I'm really sick of the Sony splash screens
  now.).
Treat this part as mandatory: Feed each chocobo 10
  sylkis greens (at least!) and race them to class A or
  S.  You could race your "wonderful" chocobo to S class
  while you're waiting for your Black chocobo to grow up.
You'll also need a Zeio nut for this one, so go get
  one if you don't have it.  Sell all your other nuts or
  throw them away unless you want to repeat this process
  later.
Now the finale; Mate your black and dashing chocobos
  with a Zeio nut.  With S class chocobos each fed 10
  Sylkis greens, many people have reported getting the
  Gold chocobo on the first try.  It still took me
  about 8 or so attempts.  You are likely to get more
  Black chocobos, or worthless yellow chocobos.  Don't
  worry, you aren't likely to mistake the Gold chocobo
  for a yellow one and reset.  The Gold is like the same
  color as the bottom part of the "S" in the Playstation
  logo splash screen.
Once you have the Gold, go get the KOTR materia!
  Naturally, the Gold chocobo is the best racer if you
  want to win stuff.  It's top speed is close to 150
  and it isn't slowed down by any area of the track.