What's the big deal about learning Abilities Anyways?
A Chocobo's abilities could be comparative to
a player's Job Abilities.  They give that extra edge to a Chocobo during Races and while digging for items.
A Chocobo racing with First-Class Strength and Endurance and featuring Gallop and Auto-Regen is going to leave a Chocobo with First-Class stats and no abilities in its dust.
A Chocobo with Burrow and Bore is going to pull in more and better items than one without these abilities.
 
Having abilities also makes it possible for future Chocobos being bred from your Chocobo's card to immediately inherit one of the abilities you have taught your Chocobo.

Deciding which Abilities are best for you:
Using the Tell your Chocobo a Story care option, your Chocobo can learn a maximum of 2 abilities.
These are learned from 5 Chocobo stories you acquire from Cut-Scenes with your VCS trainer (the NPC you visit while taking care of your chicken).  I'll explain more later, lets introduce first the various abilities you can assign to your Chocobo and their functions:
 
1.      Gallop
increases speed
(based on Strength)
2.      Canter
more efficient running but slower than Gallop
(based on Endurance)
3.      Auto-Regen
increases the rate at which energy is restored
This may also affect stamina during chocobo racing. 
Therefore, most likely based on Endurance.
4.      Bore
dig type: poking  (increased chance of acquiring more valuable items)
Ability based on Discernment.
5.      Burrow
dig type: scooping  (increased chance of digging up items)
6.      Treasure-Hunter
increases your chance of finding the digging spot in the 'Hot 'n' Cold' game.
Chocoshop's Ranch offers Abilities 1-5.

Personal Preference:  Unsure what to choose?
For a Racer, Chocoshop personally prefers
the combinations:
Gallop & Auto-Regen   -and-
Canter & Auto-Regen
For a Digger, Chocoshop strongly promotes the combination:   Bore & Burrow.

Ok, so now I know what my choices are,
but how do I learn them?
 
1.  Inheritance
Those breeding cards with the Jeuno matchmaking NPC may end up with a Chick already inheriting 1 ability.
The quality of the parent cards play a pivotal
role in whether or not your chick acquires an
ability at birth, as well as Chance.
-  parent cards with high Receptivity increase
chances of one of the abilities to be passed on.
-  both parent cards sharing the same ability
increase the chances of inheriting that ability.
-  Chance.  Keep in mind a Chocobo will not inherit an ability if neither of the parent cards feature abilities.
 
2.  Acquiring Stories
Story Key Items can be obtained by doing your Cutscenes at your Caretaker daily.
The Process is:
 
Do appropriate Cutscene ->
Obtain Story Key Item ->
Tell Story to Chocobo

Below are the names of the different Stories,
what ability you learn by telling them, and
which cutscenes you learn them from:

1.  Acquiring Gallop:
-  meet up with Hantileon in your cutscenes
3 seperate times.
-  key item:  Story of an Impatient Chocobo

2.  Acquiring Canter:
-  meet up with Brutus in your cutscenes
3 seperate times.  [long walk]
-  key item:  Story of a Curious Chocobo

3.  Acquiring Auto-Regen:
-  Win Compete Against Others 3 times.
-  key item:  Story of a Happy Chocobo

4.  Acquiring Bore:
-  meet up with Pulonono in your cutscenes
3 seperate times.
-  key item:  Story of a Youthful Chocobo

5.  Acquiring Burrow:
-  meet up with Zopago in your cutscenes
3 seperate times.
-  key item:  Story of a Worrisome Chocobo

6.  Acquiring Treasure Finder:
see:   Treasure Finder
-  easiest ability to acquire
-  complete the mini Cutscene Quest:
Lost Chick's Owner.
-  key item:  Story of a Diligent Chocobo

The amount of success associated with
learning an ability is directly linked to the state
of mind your Chocobo is in and the amount of
Discernment your Chocobo has.
 
A spoiled or restless Chocobo has a far less chance of learning that desired Ability than a happy or energetic Chocobo.
Each story requires a different amount of Discernment that the Chocobo must obtain before being able to learn the ability.
 
A great thing about Telling Stories to your Chocobo is that it actually somewhat increases your Chocobo's Discernment as well as affection each time.  So while you are trying to learn abilities, you are also helping your Chocobo's stats.
 
Because Discernment is the pivotal factor in
being able to learn a story, the Chocobo must
be following a Mental Agenda, and not a Physical Agenda during its early stages of development.
 
A Mental Agenda will focus and improve the primary Stats:  Discernment & Receptivity.
while a Physical Agenda focuses and improves the primary Stats:  Strength & Endurance.
This is the reason why those that raise a Chocobo to First Class Physical Stats and acquire 2 Abilities take more than twice as long
to build up their Chocobo than a normal Physical Chocobo without 2 Abilities.
Physical Chocobos with 2 Abilities follow a 2 Step program initially Mental, then Physical after learning 2 Abilities.
Physicals without abilities or who have inherited just 1 ability follow just the Physical Program until maxed stats.
Beginner Raisers and those that don't Check their Chocobos on a daily basis are strongly encouraged to not do this 2 Step Mental then Physical program, as it's time consuming and confusing for beginners.