Level and MSQ Boosts

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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