FEEDING

Feeding is fairly simple. Every Rank you gain, you'll get one training session to use to feed your chocobo. Training sessions will stockpile, so it's okay to save up for better food if you want to. The different grades of food raises stats at set rates. Grade 1 will raise stats by 1% of Stat Cap, Grade 2 is 2%, Grade 3 is 3%. Decimals here do not show, but they exist so you may occasional see 'bonus' stats appearing. This is simply the decimal rounding up and down.
Grade 1 Feed is available from a vendor in Bentbranch for 1500 gil. It can be crafted as well, but the materials are more expensive then just buying it from the vendor. Grade 2 Feed can be purchased for 1100 MGP, can be bought from the marketboard, or crafted from a 2* CUL recipe which requires 1* MIN & BTN mats, along with cross-bred gardening veggies. Grade 3 can only be purchased with MGP.

STAT CAPS

This formula took a little crunching to figure out. Your stat caps are determined by (40x((Pedigree-1)+Star Rating))+20. So, your starting chocobo is Pedigree 1 (P1) and has 2 Star Ratings in all stats, so all stat caps are 100. If a P2 Chocobo has all stats at 2 Stars, all stat caps are 140. The maximum stat caps are 500 with a P9 Chocobo with all 4 Stars. That's the ultimate goal.
Be warned, as chocobo's pedigree levels increase, XP required to Rank Up climbs up as well, but at least you'll have access to higher division races which yield more XP and MGP as well. Your rating will also climb quite quickly as you first start Ranking up. For example, a P4 Chocobo can easily start with over Rating of 20, which means even at Rank 1, you'll be unable to race in the Maiden Division.
BREEDING (This section has been heavily edited since the original posting, thanks to people contributing.)
When you breed chocobos, you'll need at least a chocobo that is the same pedigree as your retired chocobo. Regardless of how high of a pedigree chocobo you use, you'll only be able to increase pedigree by 1 per breeding. So, if you breed a P1 with a P1, you'll get a P2. If you breed a P1 with a P9 Cover, you'll still get a P2. Using a higher pedigree chocobo does mean a chance for better stats, and also getting a Tier III ability (which seems to be only possible through breeding right now), however they are more costly. There doesn't seem to be any downside to using a higher chocobo other then cost.
Stats are determined at random, but the system does have some crazy sort of logic to it, and this will take some explaining. When you breed, it's the chocobo's grandparents that'll determine what stats your bird will have. This is how the system works. For this, M1 is the male being breed, and F1 is the covered female, and M2 is the male child born. When M1 and F1 breed, the system looks back at their parents and grabs stats at random from the chocobo's parents. So, for M1, stats are picked randomly from both of those parents and form the MALE PEDIGREE for M2, and the five stats picked from the female side become the FEMALE PEDIGREE. These pedigrees are what M2 will use when he becomes a breeder. Now, from these pedigrees, stats are picked at random again to form the actual stats of M2.
So, while picking chocobo's to breed, it's important to know that while looking across the pedigrees, there is a 25% chance your chocobo will have any stat from the 4 grandsires/dams, and a 50% chance he'll be able to pass that stat on to future generations. Let's get an example here. While looking at Max Speed, you see the four pedigrees at MM - 2 / MF - 2 / FM - 4 / FF - 1. Since the Male pedigree has 2 Stars in both grandparents, it's a guarentee that the male pedigree will carry on the 2 Stars. For the Female pedigree, one parent has 4 Stars and the other has 1 Star. That means it's a 50/50 chance of which stat will be carried on. So, overall the new born chocobo will have a 25% chance at 4 Star, 50% chance at 2 Star, and 25% at 1 Star.
In a way, a chocobo's breeding potential is completely different then it's racing potential. It is completely possible for your new chocobo to have lower stat ratings then the sire, but higher pedigree still means higher stats as well. Abilities and colors have a 50/50 chance of being passed on, your fledgling with have the color and ability of either of the parents, there is no 'mutating'.

RETIRING CHOCOBOS

Once you have reached Rank 40, you can retire your chocobo. Current stats make no difference, only stat ratings will be passed onto the offspring. Also, your ability you have will be locked in to be passed on as well. If your chocobo has 2 abilities, you will be able to choose which one you want locked into pass onto your young. Retired Chocobos can be bred up to 10 times.

RETIRED VS COVERING CHOCOBOS

Covering chocobos will have random stats and abilities, and are ideal while you are getting your pedigree raised. However, they can only be bred once. For this reason, Retired chocobos are more ideal mates as they give much more control. Since retired chocobos can be bred 10 times, if the offspring doesn't turn out to your liking, you can simply discard the offspring and try again. However, this is more once you start reaching Pedigree 8 or 9 and are trying from your perfect chocobo.

CHOCOBO STATS AND EFFECTS There are five stats that affect your chocobo.


MAX SPEED - Simply affects how fast your chocobo can go.
ACCELERATION - How fast it takes to go from minimum speed to max speed.
ENDURANCE - How long it takes while running to enter Lathered state.
STAMINA - Maximum Stamina pool
CUNNING - AKA Balance or Traction, it's how easily you can change lanes and also how fast you can take corners without slowing down.

Now, the following negative status effects can be inflicted during battle through attack items, or by hitting panels and mobs.

HEAVY - Slows down the chocobo.
FRENZY - Makes the chocobo run at full sprint and increases Stamina Drain
SILENCED - Unable to use any active chocobo abilities.
DISTRACTED - Unable to use any items.
LAMED/EXHAUSTED - Drops chocobo to minimum speed and unable to accelerate.

BASIC CONTROLS AND RACE LAYOUTS

Controls are fairly simple for Chocobo Racing. You can accelerate, steer left and right, and jump. That's it. As you accelerate, you will start to burn through your Stamina. You'll quickly notice that there is a delay between using/losing Stamina, and it reflecting on your gauge. This is very important when trying to control your Stamina, as even if you let go of the accelerate button completely, it takes almost 10% Stamina before you stop using at all.

If you keep the accelerate button held down, you'll soon enter the Lathered state and the Stamina gauge will start to blink/glow, and suffer increased Stamina Drain. This isn't too critical, as you'll need to enter it quite often during a race, but it does require some control. If you completely run out of Stamina, you'll enter an Exhausted state and there is no way to recover from it without using items, abilities, or panels that recover some Stamina.
Through the racetrack, you'll encounter panels and item coffers. Item coffers give random items that are displayed in order before approaching the coffer. You can only hold one item at a time (unless a passive ability exists that allows more). Simply run through the coffers or jump to get the high ones. There are 4 kinds panels through the race courses as well. During Maiden division, there are no harmful panels, but every division after that has them. As of R-100, I've let to encounter any race track where you can't just simply steer around the panels, however this changes in R-120+ - Courtesy of /u/ZettaiRyouikiXIV . Panel effects are as follows.
BLUE - Grants Sprint
PURPLE - Grants Heavy
GREEN - Restores 10% Stamina
RED - Deals 10%~ Stamina

Once you reach R-60, mobs will start appearing on race courses. Each mob has a different effect, from R-60 Costa Jellyfish causing Silence and Distracted, to R-80 Tranquil Imps stealing items, to R-100 Tranquil Malboros doing a large aoe Heavy. Either way, it's always best to just avoid them outright.
Experience points vary by division, with R-40 giving 180 XP for a 1st Place, R-60 gives 210 XP, R-80 gives 240 XP, etc. etc. After every race, one racer at random will be chosen from 2nd-8th place to receive bonus XP. This Bonus regardless of ranking will give the player XP equal to between a 1st and 2nd Place finish.

ATTACK ITEM/ABILITY EFFECTS

GRAVIBALL - Causes 5s Heavy to the racer in front of you. If you're in 1st, No Effect.
BACCHUS WATER - Causes 15s Frenzy to the racer in front of you. If you're in 1st, No Effect.
BRAIR CALTROP - Creates a field of Brairs around you that deals up to 20% Stamina to anyone caught in the area of effect.
CHOCO SILENCE(A) - Creates a field of gold chains around you that causes Silence to anyone caught in the area of effect. Duration is determined by rank.
CHOCO SHOCK(A) - Creates a field of blue chains around you that causes Distracted to anyone caught in the area of effect. Duration is determined by rank.
CHOCO STEAL(A) - Has a chance to steal the item from the chocobo in front of you, the higher the tier of Steal, the better chance you have to succeed.
CHOCO METEOR - Causes 20% Stamina damage and 6s Lamed to every chocobo in front of you.

BENEFICIAL ITEMS/ABILITY EFFECTS

CHOCO POTION/CHOCO CURE(A) - Restores Stamina. Choco Potions restore 9% Stamina, Choco Cures restore 3% per tier.
SPRINT SHOES/CHOCO DASH(A) - Chocobo enters Sprint. While Sprinting, you'll achieve higher then max speed and also suffer no Stamina Drain temporarily. Sprint Shoes lasts 3s, Dash for 1s per Tier.
CHOCO ESUNA(A) - Removes status effects. Esuna has a percent chance of removing all Status Effects (except Silence because you can't use it then). The higher tier, the higher the chance of success
CHOCO CALM/EASE(A) - Calm and Ease remove Frenzy and Heavy respectively with 100% success, and also restores some Stamina. Higher tier abilities restore more Stamina.
CHOCO REFLECT(A) - Reflects back any non-aoe attack back at the caster. Duration of reflect lasts depending of tier.
CHOCO AETHER - Restores the use of your Active ability. If you have no active ability, or it hasn't been used yet, the item will be wasted. If you have two Active abilities and both have been used, only one will be restored.
STAMINA TABLET - Chocobo becomes immune to any Stamina Drain/Damage for 10s. This includes Stamina Drain from sprinting, as well as attack items like Briars. If your Chocobo is already at 0%, the item will have no effect.
HERO TONIC - Increases all stats and become immune to all enfeebling effects for 15s.

PASSIVE ABILITIES

There are multiple passives available in the game, it's a little too early to say all of them for sure. Some trigger at the beginning of the race, such as Head Start. Some are always active like Heavy Resist. Some boosts the use of items like Speedy Recovery.

ACQUIRING ABILITIES

Abilities will be acquired from Rank 10 onwards. Your chocobo will learn one ability at random. If you don't like the ability you've learnt, you can use a Lithe Water which can be purchased for 800 MGP to erase it, and the next time your Rank goes up, you'll learn another ability. You'll also get a secondary ability through breeding which can not be changed except through more breeding.

stat growth

FLOATING DECIMAL

This is the most important thing to cover right away is how the system deals with the floating decimal that is present in chocobo stats. Your stats will always be displayed rounded down to the closest integer. So, if your chocobo's stat is displayed at 20, that means it can be anywhere between 20.0-20.8. This becomes very important as everything is done as percentages of the stat maximum.
Because of this floating decimal, people started to think bonus stats might be involved while feeding because certain stats would get more points, making a Grade 1 give +2 or +3 when it should of given +1 or +2. This is simply the floating decimal showing up from time to time. For example, let's say you have a stat with a stat cap of 260. Your stat would grow like 26.0, 28.6, 31.2, 33.8, 36.4, 39.0. But, the game would display 26, 28, 31, 33, 36, 39; making it look like a bonus showed up from time to time.

STAT CAPS

This formula took a little crunching to figure out. Your stat caps are determined by (40x((Pedigree-1)+Star Rating))+20. So, your starting chocobo is Pedigree 1 (P1) and has 2 Star Ratings in all stats, so all stat caps are 100. If a P2 Chocobo has all stats at 2 Stars, all stat caps are 140. The maximum stat caps are 500 with a P9 Chocobo with all 4 Stars.

RANKING UP

When your chocobo is Rank 1, it'll have 11% of the stat cap. (Note: The starting percentage might vary based on pedigree, everyone started keeping track at P2) So, if your Chocobo's stat caps are 180/180/140/180/220, your starting stats will be 19/19/15/19/24 because of the floating decimal. That means, you have 445 total percent (or let's call them points to make it simple) to gain before being completely maxed out.
Each rank, you gain 5 percent points that are spread randomly between all the stats. That means you'll either see 5 different stats go up by 1%, or you might see 3 stats go up by 1% and 1 stat go up by 2%, or 1 stat go up by 1% and 2 stats go up by 2%. I've yet to see data of 1 stat going up by 3% from ranking up, but it is possible depending on RNG. That means, going sololy off of gaining points through Ranking up, it'd take you until Rank 90 to hit max assuming stats spread in a perfect manor.

FEEDING

Rank 90 is a LOT of racing to achieve max stats, so this is where feeding comes into play. Grade 1 food raises a stat by 1%, Grade 2 is 2%, etc. So, when we add food into the formula, the Rank required starts to fall quickly.
Assuming using only Grade 1 food, you can achieve max stats at Rank 76. Using Grade 2 food, Rank 65. And using only Grade 3 food, max stats are achieved at Rank 57. The maximum Rank is 50 at which point, your chocobo stops gaining XP. So, it seems it is impossible to achieve 100% maximum stats at this time.
Since points from Ranking up seem to be completely random, when trying to go for a maximum stat Chocobo, it'll probably be best to stop feeding around Rank 40, and letting points from Ranks fall where they will. You should NEVER feed up to maximum in one stat as you could easily start wasting points. This is completely dependant on how the RNG system works and if it'll keep trying to apply points to a stat that can no longer go any higher, of course. But to play it safe for now, I'd avoid feeding past 80% of stat cap. This is recommended, not fact.

ACQUIRING FOOD

Grade 1 food is the easiest. You can buy it from the Feathervendor in Bentbranch for 1500 gil. You can craft this food yourself, but the materials are extremely expensive and it is impossible to get HQ Feed. So, just buy your Grade 1s. Personally, that's all you should need for your first few chocobos.
Grade 2 food has multiple ways to acquire. The easiest is buying it from the vendor in Gold Saucer for 1100 MGP. If you want to try to stockpile as much Grade 2 as you can, you can easily buy the food as soon as you have more then 1100 MGP, then pay gil to get back up to 500 MGP, making it so you only have to earn 600 MGP per food (and roughly 5k gil).

Grade 2s can also be crafted. It is a 2* CUL recipe, but since HQ is impossible, you don't need any melds to craft this feed. You can simply use food to boost your crafting stats high enough to craft all the feed you need. The materials for this feed are from 1* MIN/BTN nodes located in Eastern Thanalan. The MIN node is at 9 AM, and the BTN node is at 1 PM. You can only get one mat from each node per Eorzean day, you can hit both nodes in the same day. You will also need the higher-level chocobo feed that can only be acquired through cross-breeding plants, each feed uses a different plant. FFXIV Gardening is an excellent resource for all of your gardening questions.

Grade 3 food can only be acquired through the vendor, costing 2750 MGP each. Start stocking up now if you intend on using Grade 3s.

I'm going to answer this here because it seems to be the most common question. 'Do I need to feed my chocobo if I'm just planning on breeding it?'
No. Feeding is 100% optional, however, feeding means you'll have higher stats for racing which means more victories, YAY! Choosing not to feed will save you gil and MGP, but require you to run more races to reach Rank 40. But again, feeding is 100% optional if you're only worried about breeding

RANK

Rank is pretty much your chocobo's level. That's the simpliest way to put it. As you acquire XP, your rank will increase. Every Rank increase will get you 1 Training Session and 5 Percent Points applied at random across your stats. At Rank 10, your chocobo will start acquiring racing abilities. At Rank 40, you can retire your chocobo for breeding. At Rank 50, your chocobo will stop gaining XP, meaning that Rank 50 is the max Rank.

RATING

Rating is the overall rating of your chocobo's ability, and determines what divisions you are allowed to race in.
This is the real reason I'm writting up this guide. There is an incorrect formula floating around the internet about how rating is the average of all stats. This is very incorrect. The correct formula is long, involves a lot of ugly decimals and such. This is the simplier version formula though.
(Combined Max Stats/500) x (Rank+10) = Rating

Using the correct formula, we can see that ONLY Rank and Pedigree determines your Rating. Feeding your chocobo does not affect your Rating at all, meaning that all stats gained through feeding are purely bonus and put you above your opponents. Just another reason to feed, eh.
Also, using the above formula, we can see that the MAXIMUM RATING any chocobo can obtain is 300. For this reason, I expect R-300 to be the highest racing division. So, for all those hardcore racers out there, there's a lot more to come.

285 RATING ACHIEVEMENT

Since we now know what affects Rating and what doesn't, I've figured out how to reach this achievement and get the title 'Chocobo Trainer'. Be warned, this is the hardest achievement to get in Chocobo Racing, it's 'The Final Witness' of the jockey world.

In order to get a Rating of 285, you need a P9 Chocobo with a combined Star Rating of 17/20 and achieve Rank 50. If you have more then 17 Stars, you might hit 285 at Rank 48 or 49, but 17 Stars is the bare minimum in order to get to 285 Rating.