In Final Fantasy XIV, mounts are some of the most sought-after items for players to get their hands on. While there are a few mounts that are unlocked over the course of the game's main story, those are few and far between.
Each of the Gold Saucer mounts can be purchased from the Prize Claim NPC at the main counter in the Entrance Square. They'll be listed under "Prize Exchange III". You'll need a large sum of MGP to purchase any one of them, let alone all of them. There are plenty of ways to earn it, from the various tournaments to GATEs. One of the best ways is to simply complete the Fashion Report each week. For just participating, you'll earn 10,000 MGP. For getting a score of 80 or more, you'll get 60,000 MGP in total.
While you're there, you may as well participate in the Jumbo Cactpot and the daily Mini Cactpot. It's tough to get lucky with the Jumbo Cactpot, but getting the best result in the Mini Cactpot isn't too rare, and it'll earn you 10,000 MGP. No matter how you choose to go about earning MGP, it'll take some time to save up enough for the pricier mounts on the list. For some, though, the grind is worth it.