Personally, I've been leveling the dragoon (2nd job past 50) using the daily Clan Hunts [level 1 and 2] plus that Vanu beast tribe quests. Kill 1-3 beasts five times for each level of clan hunt, get around 125K XP. Do 3 beast tribe quests (they actually adjust for anything level 50+), earn another 150K XP.

If you have NOT reached level 60 on anything yet, do the quest lines and dungeons that accompany them. I reached Azys Lla and got to ride the dragon just before hitting 60 -- and finished the MSQ after spending a small amount of time finishing the Clan Hunts for level 1, 2 and 3 that day.

Note that you might not get any level 60 dungeon instances if you don't complete the MSQ. You certainly can't fly in an area until you've finished the last quest there (which is usually an instance unlock and run).

Good luck.