Before you can ride a Chocobo you will need a Chocobo raising license.
You can obtain this from the Chocobo stable in Jeuno.

You must be Level 20 in your Primary Job before you can acquire a
raising license. Once you reach this level though follow the steps
below:

Firstly you will need to get Grass from Wadi Hare located in Wadi. You
will need four of these.

Next you must go to Jeuno. This is the only Chocobo stable which gives
out the licenses. If your Primary Job Level is in the low 20's getting
there isn't easy. If you can find a high level character to escort you
there in a party it will help a lot. Otherwise you may need to wait
until you are closer to your 30's to 40's in your Primary Job before
attempting the trip.

After getting to Jeuno, look at the map and and head to the Chocobo
Stable. Once you arrive there talk to Brutus.

Brutus will ask you two questions, for the first question choose the
first answer, and on the second question choose the second answer.

Now talk to Osker, who is close by, and Osker will tell you to go and
get the four grass you should already have obtained from Wadi. Choose
"trade" from the menu and select the Chocobo that is right next to Osker.
Put one of the Grass into the trade box and click the trade button. The
Chocobo will make a sound.

The first and second time you do this the Chocobo will not eat the
Grass. Take note of the time in the bottom left of your screen. You can
make one attempt per day (it takes 6 game days in total). After the
first two times you give the Chocobo the grass it will start to eat it.
So the four grasses will be consumed on the 3rd, 4th, 5th and 6th tries.
After you have fed it all four grasses you will obtain your license and
can start riding Chocobo's!