It is very rare that your stats would go up from just walking your chocobo. That really only raises affection. You will find that raised chocobo's are slow - slower than rented ones, unless their strength/endurance are first-class. In order to get them to first class, you have to have a very rigorous training schedule - taking a walk in town,a nd carrying packages until both attributes get to first class. Here is where it gets tricky - unless you are feeding good greens and (along with carrots), your affection level will plummet.

If you are raising a chocobo not for riding or racing, but rather for breeding or digging, you want a different plan, i.e. listening to music, etc, to raise discernment/receptivity, so you can teach it some abilities to use for digging (burrow/bore), or abilities you want to pass on to your next generation of chocobos (cantor/gallop).