Level and MSQ Boosts

[Jojoya]

In your situation, FATEs will be the only decent source of open world XP for your chocobo since there's not much other open world content that will level sync. HW beast tribe quests don't include that many kill creature quests so those are of little help. I'd focus on FATEs that spawn large numbers of creatures to kill until you get your Bosom Buddies challenge log entries completed for the week. Feed your chocobo a Curiel Root after you summon it for a 10% XP bonus, or 15% if it's the preferred food (see stable training below).

The other way to level your chocobo is stable training. If your FC has a house with a stable, make certain to stable your chocobo when you're not doing FATE grinds. Not only are you able to train your chocobo once an hour, other members of your FC can do it once an hour (starting an hour after you stable it). If your FC doesn't have a house, I'd suggest getting an apartment and using the apartment stable since other tenants in the same apartment build can train your chocobo if they are willing. Feeding your chocobo the same buff food (Curiel Root, Tantalplant, Mimett Gourd, Sylkis Bud, Pahsana Fruit) when stable training will eventually set its preferred food for an additional stat or XP bonus. If you do choose Curiel Roots, I would suggest using Krakka Roots for further stable training once the preferred food is set since Curiel Roots can be expensive on the MB.

Remember that stable condition affects how much XP chocobos will get when being trained - you always want stable condition to be Good for max XP benefit. Magicked Stable Brooms are usually cheaper on the MB than on the vendor (20k gil each at the vendor) and their cleaning effect lasts for about 3 days.

You will be doing fates and beast tribe quests for gems and rewards at SHB you will do it for riding maps and special mounts.. You can level chocobo at fate or beast tribe quests.
Chocobo will get a boost at weekly "achievement" Bosom Buddies I / II (Challenge gives %5 and %10 of max xp) you can repeat it every week I believe.

Note: Despite using chocobo at every opportunity mine still rank 12 :P

Once your character hits level cap, best source of open world XP for chocobos is grinding random mobs in the Tempest. Most of the FATE mobs are level 76 and give about 4k XP while the random level 79 mobs will give 6-7k XP. I've been working on leveling my alt's chocobo lately (it's also 12 right now) and it goes fairly well.

For players who are in the 63-70 level range, try to get a group formed to do the 13 part Dataqi Chronicles FATE chain in Azim Steppe. More than enough mobs spawn over the course of the FATE chain to get both challenge entries completed. Actually just did it again last night for old times sake with others in the hunt community and was getting about 500 XP per mob kill, which isn't that much compared to what I get in Tempest but the sheer number of mobs that were spawning with 2 full parties working on the FATEs together helped make up some of the difference.