To give your chocobo exp while it is in the stable, you must feed it krakka roots. The amount of exp you get per training session varies according to the bird's level. You cannot feed it once it hits max exp for that level; so, once you max level 8, you need to remove your bird from the stable and give it exp once more while fighting something in the open world. It will then level up to 9 and you can put it back into the stable for more training. You can feed the bird once per hour.

Thavnairian Onions and other rare bird food have been grown from gardens. You can find that information here in the Items etc. subforum:
http://forum.square-enix.com/ffxiv/t...ing-Harvesting