Level and MSQ Boosts

I created a level 1 character and then got and used a level 70 boost. On other characters I have done the quest grind several times that gets you to Ishgard and I didn’t want to do the main stories from level 1 to that point on my new level 70 so I also got a main story skip. I got the lesser one that completed all my main stories to the point where I had the quest to enter into Ishgard. I didn’t mind doing the stories from that point on, which includes doing the aether currents for flying.

Even though I could start using my mounts on the ground immediately I did have to do the feisty chocobo quest to unlock him as a companion; in the past I’ve done that on my other characters in their early 20’s. The problem is that the chocobo companion rank is 0 with no skill points. Since I am now doing level 50 main story quests at level 70, I get very little exp and therefore so does the chocobo. Usually by level 70, on my other characters, he would have several ranks by then. On my now new 70, being 20 levels above the MSQ’s, I don’t need the help of the chocobo but I thought it was strange that after the Level and MSQ boosts, that the chocobo rank level was kept at zero. My new 70 is and will be doing the daily roulettes and with the exp from the challenge log which resets weekly, I may be lucky if my chocobo reaches rank 1 by the time I get to 80. I didn't see a chocobo boost on the Mog Station, unless I missed something?

There's no chocobo boost.
If you only intend to play that one job, then you're probably out of luck. You'll also be level 80 by the end of the HW content if you keep doing daily roulettes as well.
So you'll probably end up leveling a second job anyway if you want playing that character to be at all interesting.
Now I get that you don't want to level from scratch, so you have 4 other options available; Red Mage, Samurai, Gunbreaker and Dancer.
Pick RDM or SAM and you'll be progressing through the story at a relative level, and getting the optimal EXP from fighting in the overworld, thus levelling your chocobo.
Your best bet though, is to stable it and feed it every day. It'll still be slow, and you'll need to take him out every now and then to actually rank up, but it's better than nothing.
Or, as it's an alt, don't even bother with the chocobo.

No, there is no boost for your chocobo - its also quite normal that the bird gets very little exp from just killing mob in the overworld, even if you're at level.

There are two actual good sources for chocobo-exp:
The first is the weekly agenda bonus that you get for killing a certain amount of mobs with your bird by your side - my guess would be that thats what got your main characters chocobo up to a decent level while playing through the MSQ: A couple of weeks of weekly boni.
The second one is placing your bird in a stable at your FC-house (if you have one) or at your apartment-building (if you have an apartment) and feed it with Krakka-roots - other members of your FC can feed him aswell. There is a one hour cooldown on feeding the chocobo and even if you're offline other people in the FC might take care of him!

At the end of the day leveling the chocobo is quite a slow grind with no huge benefits (yes, its nice that your chocobo can heal you, but its not a dealbreaker) and as you said it yourself: you dont need his help right now. With the weekly bonus you'll get there soon enough again.

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A couple of weeks of weekly boni.

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The plural of bonus is bonuses. ;)

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

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.

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Originally Posted by Insertusernamehere

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

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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.

Not sure how terribly chocobo leveling with blue mage was nerfed recently, but I was averaging ~20 minutes a choco level from 12-20 ae'ing in North Shroud at the end of SB on alts.