best way to level chocobo

[Penthea]

You can buy a variety of chocobo food from the Vath beast tribe: https://ffxiv.gamerescape.com/wiki/Vath_Stickpeddler

[Moomba33]

I never had the patience to straight up grind mobs for my bird.
I got my chocobo to rank 20 during Heavensward through a combination of the Challenge Log, feeding Krakka Roots, and just making sure it was always summoned when I was doing something in the overworld (quests, clan hunts, FATEs, treasure map parties, ect).