Yes, as the others have said, you can train your chocobo every time you level up. Each level, some stats go up seemingly randomly, but you earn training sessions with every rank up. These are used by buying feed from the trainer, to increase either of the five attributes. And the higher Pedigree (overall level, 1 - 9) of your choco, the higher the max level of those stats.

You already know about breeding since you're 4, but if you haven't, put some thought into the "stars" of the GRANDPARENTS (each mating pair displays what its parents' stars were in each stat). Those are factored into what the offspring's stats (stars) will be.

Choco racing is for the long haul..