Basically, the Chocobo Subquest is a longwinded way to "leveling up" your chocobo to access secret areas and gain certain secret items. As your chocobo level's up, it will also gain new abilities and change colors.

Light Blue - Lets you search shallow water on the beaches.
Red - Allows you to climb over any mountain.
Dark Blue - Lets you go right into the sea and search.
Gold - Allows you to fly(forests must be used as take off points).

To play this illustrious "Chocobo Hot and Cold Game" you must find one of the three game playing areas. The first "Chocobo Forest" is to the east of Lindblum. You can first access this area on Disc One, the second in a marsh found(?). The Third is the "Chocobo Air Gardens". The third one is trick to find as it's location is basically a shadow which changes location everyone once and a while(only appears in 5 places).

Enter the forest, pay the moogle 60 Gil and you can dig for treasure for 60 seconds. Ride your Chocobo and press Square to dig. Your Choco will make sounds depending on how far the treasure is from that position:

For finding these treasures you gain points and you get to keep the item you found for your self. Usually the items are nothing but Annoyntments and other very common items, other times you'll run into Gil, Dead Peppers and ChocoGraphs(or Chocograph pieces). The more you play and the more you find, the more ChocoGraphs/Pieces you find, your chocobo's Beak also levels up. Some times the items are found under ground and you must repeatedly tap Square to dig for them, the higher the level of your chocobo's beak - the faster you dig up these items.

ChocoGraphs are like a map, that you use to navigate the outside world to find treasures, the treasures found in the outside world are much better than the ones found playing the Chocobo hot and Cold game. In the outside world is also where you find chests full of "mist" which levelup/change the color/abilities of your chocobo. Once your chocobo has leveled up you can go back to previous places where you played the chocobo hot and cold game and exploit the chocobo's new abilities to search new areas previously unreachable. ChocoGraph Pieces are clues to the Chocobo Air Gardens and it's location, other than that they don't serve much purpose.

Eventually you will stop finding chocographs and your chocobo will have leveled up until it is gold. From here you'll need one dead pepper and you'll have to locate a dark shadow. Fly under the shadow and use a dead pepper to gain entrance to the Chocobo Air gardens where you will find the remaining Chocographs and Ozma(Hard).

In the outside world, when you find chests they are packed with rare items and cards, make sure you have under 100 cards when you find a chest - if you have 100 cards and you find a rare one(some can ONLY be gained from chests and can ONLY be collected once) or the new RARE one will be DISCARDED and lost forever(bad thing).

To find the chests in the outside world look for Cracks in the walls or bubbles in the sea, at these places use a dead pepper.

Name	Picture Location	Description of Location	Rewards
Streamside		Located Southwest of Chocobo Forest in the location where the river meets the beach.	2 Elixirs, 3 Hi-Potions, 4 Ethers and 2 Germinas Boots
Between Mountains		Located west of Dragon's Gate in between the mountains. (Loop around the big one)	5 Potions, 5 Hi-Potions, 2 Tents and 2 Cotton Robes
Healing Shore		Located northwest of Cleyra.	Upgrade to Light Blue
Uncultivated Land		Located south of Ice Cavern (Need Airship)	1 Jade Armlet, 3 Wing Edge and 1 Cargo Ship Card
Abandoned Beach		Located east of Black Mage Village or due west from Earth Shrine.	9 Phoenix Pinions, 5 Phoenix Downs, 13 Peridots and 1 Diamond Glove
Cold Field		Located east of Esto Gaza.	5 Echo Screen, 7 Hi-Potions, 3 Tents and 1 Theatre Ship (Prima Vista) Card
Forgotten Lagoon		Located northwest of Qu's Marsh on Forgotten Continent near Ipsen's Castle.	8 Gysahl Greens, 5 Ethers, 7 Hi-Potions and 1 Dragon Claw
Faraway Lagoon		Located northwest of Oeilvert, but you must follow the shallows from near Chocobo's Lagoon to loop ALL the way around to the dig site.	37 Potions, 6 Magic Tags, 1 Shield Armor, 1 Gaia Gear
Abandoned Lagoon		Located southwest of Fossil Roo, but you must go southeast and follow the shores in a western direction when you hit the beach.	6 Soft, 4 Ethers, 1 Leather Boots and 1 N-Kai Armlet
Bird's Eye Lagoon		Located northwest of the western beach near Lindblum. Follow the shallows as far out as you can go.	8 Potions, 4 Phoenix Downs, 3 Ethers and 1 Magician's Robe
Small Beach		Located southeast of Lindblum. Head to the eastern beach and enter the shallows until you are directly south of the Chocobo Forest.	4 Remedies, 2 Elixirs, 8 Rising Suns, 1 Oak Staff
Dawn Lagoon		Located on the southeastern Penninsula of the Mist Continent.	Upgrade to Red
Forbidden Forest		Located where the mountains that surround the Burmecian, Alexandrian and Lindblum nations. (The center of the Mist Continent)	7 Ethers, 2 Elixirs, 10 Wing Edges, 1 High Mage Staff
Green Plains		Located northwest of Alexandria. Follow the cliff in which Alexandria lies on to the point where you reach the mountain and go over it.	Upgrade to Blue
Dusk Plains		Located in the Seaways Canyon. Enter the canyon and reach a point where you are southwest of Oeilvert and north of Daguerreo.	12 Phoenix Downs, 14 Ores, 1 Kaiser Knuckle, 1 Iron Man Card
Forgotten Plains		Located due south of Ipsen's Castle.	17 Ores, 5 Ethers, 14 Opals and 1 Demon's Mail
Sea At Dusk		Located northeast of Oeilvert. From Oeilvert, follow the canyon and when you reach the intersection, continue eastbound until you hit the beach. Begin digging when you hit the waters.	15 Phoenix Downs, 1 White Robe, 1 Diamond and 1 Masamune Card
Ocean		Located northwest of the Archipelago's Qu's Marsh and west of Oeilvert. There is an island belonging to the penninsula that is near these coordinates. Go slightly west off those shores. You'll find it there.	27 Ores, 1 Light Robe, 1 Whale Whisker and 1 Alexander Card
Cold Lagoon		Located southwest of the Fire Shrine on the Lost Continent. It's the cove.	11 Peridots, 9 Opals, 15 Sapphires, and 19 Topazes
Mist Ocean		Located straight north of Alexandria Docks.	Upgrade to Gold
Outer Island (1)		Located on the island northeast of the Desert Palace.	21 Amethysts, 16 Garnets, 1 Genji Armor and 1 Ragnarok
Outer Island (2)		Located on the island northwest of the Desert Palace.	11 Sapphires, 1 Circlet, 1 Pumice Piece and 1 Hilda Garde 3 Card
Fairy Island		Located on the island southwest of the Iifa Tree.	33 Potions, 15 Annoyntments, 1 Holy Miter and 1 Dark Matter Card
Forgotten Island		Located east of the Forgotten Continent's Qu's Marsh.	1 Ribbon, 1 Rebirth Ring, 13 Amethysts and 1 Ark Card

Crack One is near Oeilvert. It is straight north of it in Seaways Canyon
Crack Two is near the Fire Shrine on the Lost Continent
Crack Three is not far from Crack One. It is near Oeilvert.

Dive Site 1 is due north of the most northern tip of the Mist Continent
Dive Site 2 is east of the Daguerreo Island and northwest of Chocobo's Lagoon
Dive Site 3 is exactly in the middle of a course from the most northeastern tip of the Lost Continent and the northwester tip of the Mognet Island
Dive Site 4 is southeast of Esto Gaza in the area near Shimmering Island (THIS IS THE MOST IMPORTANT ONE AS ZIDANE'S ULTIMATE WEAPON IS HERE)
Dive Site 5 is not exactly a dive site but it is Quan's Dwelling. Return to Quan's Dwelling with Vivi and Quina on Disc Three or Four and use a Dead Pepper to fish. A chest will emerge.

This should be all you need to know about the Chocobo Hot and Cold subgame/Sidequest.