Do you have the unicorn already? You get it by levling conjurer up to 30, that will open up a special questline.

The Gold Saucer currently has three mounts aswell - rangging from 200k MGP to 1 Million MGP in costs. Takes a bit of a grind, but its easy enough! Daily mini-cacpot and weekly challenge log are your friends here!

The Zu-Mount is brought back by airships - either ask your FC-leader if they can try to get you one (comes from sector 24) or maybe buy it on the market board...? At least on my server they're not thaaaat expensive anymore.

If you want to truely collect mounts, I recommend getting into PvP and earn yourself a Garo-title! You do that by buying a full set of Garo-gear and then talk to that priestess-NPC again. If you win PvP-battles with that title on you earn achievments unlocking three mounts aswell. However the priestess will not be around forever, so better make sure to get a title now!

If you can find a partner and are willing to pay some real money for it, you can always get married aswell and get a two-seater-wedding-chocobo (have to do the Gold or Platinum-plan for that).