Every Chocobo goes through, what I consider,
6 Stages of Age Development. 
Every site seems to use its own particular variation of this system, but mines the best *^^*.

Stage 1:   Egg
Chocobo spends Day 1-3 in a dormant mode directly after trading the Egg to the VCS caretaker.  During this period of time, you cannot put the Chocobo on a Careplan, and
the only sort of attention you can give the
unborn Chick is just watching it.
A few tests have shown that watching the Egg 100 times should place your Chick at Regards as Parent by the time it hatches.
This may have been modified, though, since I last checked.
 
Stage 2:   Adolescence
At Day 4, your egg hatches and an adorable baby pops out.  The Adolescent years run from Day 4-18.  During this period of time, you may read stories to your Chick, or bring it out on Short Walks.
 
You may set your Chick on a couple different Careplans now:
 
Taking a Walk:  Physical Careplan increasing Strength and Endurance, while lowering Discernment and Receptivity.
Listening to Music:  Mental Careplan increasing Discernment and Receptivity, while lowering Strength and Endurance.
 
Starting on Day 5, you need to start feeding your Chocobo.  Pastes, Carrots, and Greens are all possible kinds of food you may choose from.  Check out the Food Page to see your choices and how they affect stats and affection.
 
Stage 3:   Teen
Now the Stage that everyone can't wait for: the Teen Stage, where you can finally see what color your Chocobo is!  Tail feathers indicate the Chocobo's color.  Most of the Colors are easy to distinguish.  Some monitors may have settings on too dark, making it hard to distinguish between black and blue tailfeathers.
White endpoints mean the Chocobo will be yellow.
This Stage begins on Day 19.
 
You will see many new Careplans appearing in the menu:
Exercising Alone:  Increasing just Strength, while lowering Discernment and Receptivity.
Exercising in a group:  Increasing just Endurance, while lowering Discernment and Receptivity.
Playing with Children:  Increasing Discernment, while lowering Strength and Endurance.
Playing with Chocobos:  Increasing Receptivity, while decreasing Strength and Endurance.
Carrying Packages:  Increasing Strength and Endurance, while lowering Discernment and Receptivity.
Exhibiting to the Public:  Increasing Discernment and Receptivity, while lowering Strength and Endurance.
* I recommend one of the last 2 Careplan options as they provide the most noticeable attribute increases.  In fact, I use them all through Adulthood as well.
 
You may also bring your Chocobo on Medium Walks and Race them in your Caretaking Cutscenes as well.  Medium Walks will increase affection, while the Races will help out your chocobo's Strength.  Winning 3 Races also provides you with a new Story to learn Auto-Regen.
 
Stage 4:   Adulthood
At Day 29, your Chocobo is officially an Adult.
And what's so cool about that? Now you can
call it and ride it around!
Plus it's this stage where you will see if your Chocobo has any additional Aesthetic Features.
If Discernment is at least Average by Day 29, your Chocobo acquires a Raptor Beak as well as elongated Head Feathers.
If Strength is at least Average by Day 29, you'll notice bigger feet and fuller tail plumage.
 
The last couple Careplans are also finally released as well:
Delivering Messages:  Greatly improves Strength, while lowering Receptivity.
Digging for Treasure:  Greatly improves Discernment, while lowering Endurance.
Acting in a Play:  Greatly improves Receptivity, while lowering Strength.
* These 3 Careplans are designed to focus on 1 particular stat.  The boost to that one stat is quite noticeable, however there is a large negative affect on Affection.

Stage 5:   Senior
Your Chocobo becomes a Senior at day 64, resulting in a drastic decay of Stat Improvement.  It is possible to still increase stats, but I'd like to stress the Increase is Incredibly lowered.  At around Day 90, your Chocobo completely loses the ability to increase stats, unless you use special
Chocobo Buck Stat Programs.
 
Stage 6:   Retirement
At Day 129, your Chocobo has entered Auto-Retirement and you can no longer look after it
at your local Stable.  Keep in mind, you can actually retire your Chocobo whenever you'd like.  It's just Day 129 is the maximum age
your Chocobo can reach.
 
Upon Retirement, you receive a Plaque for
your Mog House, and a VCS Card.  You may still call your Chocobo like you normally do with your whistle.  Your VCS Card allows you to assign that particular Chocobo to your whistle
if you have more than one.  You can also print
out Chocobo Cards for breeding purposes with trading this VCS Card to your VCS caretaker.
You may also use the Card to choose which Chocobo you want to Race (in Chocobo Races). It's very important for you to keep this Card in a special location, because once you lose it, you can't recover it.

Progression of Attribute Advancement
During your Chocobo's Development, you'll
(hopefully) see an improvement in its Stats.
Food, the type of Careplan System it uses, taking your Chocobo on Walks and Keeping its Affection high, Competing it in Races, Reading it Stories, even your Honeymoon Package - all directly impact the tempo at which your Chocobo's Attributes Increase.
 
Below is the Progression of Stats and how many points are needed to obtain this level.
Explaining the Point system becomes very technical and long-winded so I'll just, for the time being, explain that every food gives a certain amount of Stat Points, as well as a Careplan followed and a Race done.
For Instance, Celerity Salad assigns +4 points to Discernment only.  Vomp Carrots assign +1 points to both Endurance and Strength, while Zegham Carrots assign +1 points to Discernment and Receptivity.

Poor:  0-31
Substandard:  32-63
A bit deficient:  64-95
Average:  96-127
Better than average: 128-159
Impressive: 160-191
Outstanding: 192-223
First-class:  224-255

Each Attribute is assigned one of the above levels, therefore a normal situation could be as follows.
Strength:         Better than average
Endurance:      Impressive
Discernment:    Poor
Receptivity:      Poor

I have been able to advance from Average onward in an ideal situation between stat levels every 4 - 5 days.  I have noticed there seems to be a minimal cap of time that your Chocobo is able to advance to its next Stat level.  This is most likely due to the fact its impossible to acquire a certain amount of points in a certain amount of time.  But it also seems to me, there is an actual minimal time assigned to cap advancement.
This would surely prevent the Chocobo from finishing a program much faster than SE's protocol calls for.

First-class is the highest Stat level your Chocobo can achieve, and may acquire it in 2 separate stats.  There is an additional buffer after acquiring First-Class that your Chocobo can attain as well.


Because the progression of Attribute advancement depends mainly on the
1.  Careplan system you follow
as well as
2.  The Food it consumes
I highly recommend you check out these
relevant pages.