(open for business!)
I'll keep this as concise as I can, everything in here will be based on my interpretation of my current research, other players findings and my own speculation.
To start with I'll explain the requirements for breeding:
1. You must have a rank 40 chocobo or higher. Remember it's rank not rating!
2. A covering that can be purchased from the tac and feed vendor at the chocobo square. Make sure you buy a covering that is of opposite gender to the chocobo you want to breed. If you have a male chocobo you need an F covering and for a female chocobo you need an M covering.
3. 6 hours are required to pass once you begin a covering, so prepare yourself for the wait. My advice would be to buy another fledgling chocobo to race in the meantime to earn some mgp and another chocobo.
Next i'll cover what we know about every piece of information you can find in the chocobo menu.
Chocobo genetics:
There are 3 main things to keep in mind here.
1. Star rating (1-5)
2. Pedigree value (1-9)
3. Acquired and hereditary abilities.
First I'll explain what the star rating is and how it affects stats.
Each chocobo has a star rating for every stat, the higher the star rating the higher the cap on any given stat e.g. 1 star gives you a maximum stat cap of 100, 2 stars gives you 140 so on so forth. I'm not 100% sure how much the cap raises per star after 2 stars but I would assume 5 stars is equal to a cap of 200 or higher.
Bear in mind that your very first chocobo has all its stats capped at 100 and all its stars at 2, this is of course contradictory to what I just said though. However I think the way it works is that your pedigree level has to be raised in order for the same amount of stars to provide a higher cap e.g. a pedigree 1 chocobo with a 2 star rating has a cap of 100 but a pedigree 2 chocobo with a 2 star rating has a cap of 140 and with 1 star that same chocobo gets a cap of 100.
Lastly star rating affects the stat values your child chocobo starts with, a 1 star rating gives your chocobo 11 stat points in the corresponding parameter and a 5 star rating gives you 24 points e.g. if your child chocobo has 1 star in maximum speed it will start with 11 stat points in maximum speed and if it has 5 stars in maximum speed it will start with 24 stat points in maximum speed. Star rating does not affect how many points a parameter gains on ranking up, but there is a possibility that it can change how many stat points you gain from training. I honestly have no idea what determines stat gain values from ranking up, it seems random.
pedigree value:
As far as I'm aware I don think it is possible to breed a chocobo that has a higher pedigree value than your lowest pedigree of the two chocobos you want to breed by a value of 1.
As an example, if we take a pedigree 1 chocobo and a pedigree 4 chocobo the highest pedigree you will get from the child chocobo is pedigree 2 because the lower pedigree value of the parents is 1 and you can only gain one pedigree value each time you breed (as far as I currently know.)
I think it's safe to say that a higher grade covering (which is the same thing as the pedigree value) has a better chance of attaining a greater overall star rating than a lower grade covering when you first use it. However you would still only increase your pedigree value by 1 regardless of the covering grade you purchased.
I can't be 100% sure on this yet but it seems like the logical conclusion from my own experience and others.
This next part has only had a small amount of research put into it so far, but I believe the relation between star rating and pedigree is as follows: For every pedigree value your chocobo has there is an increase to the maximum stat cap for each star e.g. I've already observed that a pedigree 2 chocobo has a stat cap of 140 with 2 stars while a pedigree 1 chocobo has a stat cap of 100 with 2 stars.
chocobo abilities:
You can have two abilities at max, a hereditary one and an acquired one.
When ready to breed a chocobo and purchase a covering the chocobo that is generated from that covering will have a random ability. This can be passed onto your child chocobo as a hereditary ability.
Upon retiring your chocobo you will be asked to give him/her some parting words, there are two options; one allows your chocobo to forget its current ability and the other allows them to keep it so if you want to get the covering chocobo's ability you'll need to choose the option that makes yours forget. If you want to keep your ability choose the option that makes it remember. However it will still be a 50/50 chance on which ability is passed on.
In the next section I will cover (presumably) how the star rating, abilities and colour of your child chocobo is determined. We don't know any exact values yet so we can only base it on logic with the available information.
Chocobo hand me downs:
Now that you've handed in your covering and selected the race chocobo you want to use at the bentbranch stables it's time for the two chosen chocobo's to do their business. So what exactly is handed down from the parents to the child?
First of all I will mention that your race chocobo's current stat values do not affect the child chocobo in any way that we can see. So if you have 100/100 in maximum speed this does not mean your child chocobo will naturally have a higher maximum speed stat upon birth. The only thing that gets handed down from the parents to the child are star ratings, abilities and colour (possibly preferred weather as well.)
Next, how do we determine which stars, colour and ability get handed down? Well both parent chocobos have their own individual combination of star ratings that can be passed on to the child chocobo and the number of chocobos that take part in this breeding process is two. With that in mind I feel it is reasonable to say that the determining factor is 50% given that there are only two parties involved. The way this works for star rating is that each star rating from both parents has a 50/50 chance to become the star rating for that particular parameter of the child chocobo. However I think there might be some weight behind the idea that the greater the difference between the pedigree value of both chocobos the more in favor of the lower pedigree chocobo that percentage will be e.g. breeding a pedigree 1 chocobo and a pedigree 9 chocobo will tip the balance towards the pedigree 1's star ratings.
As for colour I think it's safe to say that there is a 50/50 chance regardless of pedigree due to there being only two determining factors, it's either going to be one or the other. Also colour does not affect the outcome of a race so there should be no reason to use anything other than a 50% value.
Abilities may be a different story though as they do affect races, so it is likely that the higher valued ability will be given a lower chance of passing on to the child chocobo.
End.
I can't think of anything else to add to this 101 for now so let's say it's in an 'Open beta' stage. If there is anything you think I missed or could explain better or if you disagree with something please feel free to let me know. I welcome all discussion on this topic too as there is much more to learn as yet!
Hopefully this cleared up the whole breeding process for some people and I appreciate you taking the the time to read my thoughts and findings on the subject!