Though it never had a mini-game attached to it, I've got to say IV was the starter. VII brought them back to the forefront and IX followed in VII's place.

The reason I choose IV is because they actually did more than just transportation, it was the first to feature different colored variations, and for most non-Japanese fans, its the game that introduced the iconic "Fat Chocobo" (Sorry FFIII ).

Yellow, White, Black, and Fat. Its the first game I actually think of for Chocobos.