Personally, my favorite way of raising the chocobo is the FFIX way. FFVII's was way to complicated with needing the Great or Good Chocobo, having to deal with that Chocobo Sage loser, having to race your chocobo to get it in Rank A and having to get the specific kind of nut.
IX's was much simpler and cooler. Look for cool treasures. Even if you don't upgrade the chocobo, you get Elixirs and Cards and alot of cool stuff for the time you wasted on it. Unless your very unlucky and it takes you hours to find the Chocographs, you really can't complain. Besides, in the end, the FFIX Gold Chocobo can FLY!
I absolutely preferred FF9's Chocobo Hot and Cold Game over the Racing/Breeding game from FF7. The FF7 game was unnecessarily repetitive and yielded only a few genuinely awesome rewards. As opposed to the FF9 system which was fun, engaging and offered most all of the final weapons for the characters and a bunch of other badass equipment in the process. I just found exploring the world map and digging up treasure a hell of a lot more entertaining than racing the same damned track over and over and over again for hours on end.
I also preferred FF8's Chocobo Forest minigames a lot more interesting than FF7's breeding game.
Linear Mode
