There's quite a few activities that you can engage in with your Chocobo.  I will attempt to list these activities below, although it's very
possible that I could be leaving out a few.

1.   Chocobo Digging
see:   Categories:  Chocobo Digging
Digging is one of the popular reasons why people raise their own Chocobo.  A lot of valuable items can be dug up when your Chocobo acquires Burrow and Bore.
-   These abilities allow your Chocobo to dig up items unable to be dug up without these
abilities as well as
-   increase your chances for findings and the
amount of findings dug up is increased as well.
see also:   Mental Chocobos

2.   Hot-and-Cold Game
see:   Categories:  Hot and Cold Game
for an extensive overview of what this game is about, how it works and what you can win.

Brief Overview:
This game is an alternative means of digging up treasure without having to conventionally dig.
You can do this game 1ce every Vandiel Day.
This game offers some rather worthwhile prizes, and with a little luck can bring you a bit of extra cash.
 
Visit your VCS representative to receive a chocograph featuring several options where
you can play this game.  You will receive also a clump of Tandjana wildgrass, which is used to call your chocobo to the field.
 
You must have a registered Chocobo before participating in this event.
You must possess a map of the area you choose to hunt for your treasure at.

General Instructions:
1.  Commence your Hot-and-Cold session by
using a Tandjana Wildgrass in one of the areas featured on your Chocobo.  This calls your Chocobo to you and you may freely roam throughout this zone on it.  You are not able to pass through other zones, however.
2.  Look for a marker (???) on your map, which
indicates the approximate location of a hidden treasure chest.
3.  Ride to the area close to the marker and try digging.  Your Chocobo will provide you with messages indicating if you are close to the treasure or not.
4.  When the Chocobo notifies you that you are on top of the treasure, keep digging until you succeed in digging it up.  Note:  Each time your Chocobo Digs, it becomes more tired and at a certain point it will refuse to dig anymore.
Therefore, do your best to try and efficiently find the treasure in as little digs as possible.

3.   Chocobo Racing
see:   Categories:  Chocobo Racing
This Section contains a lengthy and deeply involved synopsis about Chocobo Races and how they work.
 
Brief Overview:
Chocobo Races via the CRA (Chocobo Racing Association) racetrack as well as in the Chocobo Arena are two ways to show off your Chocobo's Speed and Stamina.
 
The whole goal of these Races is to either win Regional Ranks for glory and a title, win gil, or win VCS Chocobucks, which can be exchanged later for nice prizes which aid in breeding (like Carrots, Greens, Pastes, medicines, Tornado Salad, Chocolixirs etc.

4,.  Camp Kweh
This is a place accessible for Raisers to visit via an NPC conveniently located outside their local stable.
In order to enter this camp, you must have a Chocobo currently residing in the Stables and at least at day 4.  You must also be in party with at least 1 other person who also has a Chocobo currently being raised.
The maximum number of people who can
enter the camp with you is 6 [Full Party].
You can view each others Chocobos there, as well as their Stats and Conditions.
There's also an option there to view what your hypothetical offspring with a particular other Chocobo may look like if bred.

5.  General Transportation
Chocobos are just one of the many means of Transportation in Final Fantasy XI.  They can be called within most open wild zones to allow a faster means of transportation than by foot.

You also avoid aggroing wild beasts in the field while Riding a Chocobo.

Chocobos can be rented in every home village as well as other large villages and small posts throughout Vandiel, and at Crags.  The Rental fee is minimal, and varies depending on the number of people renting a Chocobo there at the moment.

You may also Call a Chocobo using a Whistle you obtained during Chocobo Raising.