Final Fantasy XIV offers a wide selection of mounts to its players, and some of them take a considerable amount of work to obtain. The Magicked Card mount is one that you won't find any shortcuts for; the only way to get it is to put in the effort. For those who don't like playing Triple Triad, this one can be a real pain to get.
On paper, it's simple: to get the Magicked Card, you just need to obtain every Triple Triad card from number one through 312. The tough part is that they have varying requirements for obtaining them. Some are purchased from the Gold Saucer, some are won off of NPCs throughout the world, and some can only be gained randomly from duties. It can be tricky to track down where each card can be found, and even then, it can be a painstaking process to earn some of them.

To check which cards you've already unlocked, you'll want to open the Character menu, then click Gold Saucer, and finally Card List. Here, you'll see every Triple Triad card listed in order of number, and cards you don't have yet will be marked with a question mark. Since you only need every card up to 312 for the mount, you'll just be collecting up to halfway through page 11. Cards won by winning Triple Triad Tournaments (the cards featuring other FF protagonists) aren't included.
To avoid looking up every single card you're missing, you may want to use a tracking tool such as this one. It allows you to put in your current list of cards, and shows you the method for obtaining each one you've yet to get. The process of collecting them all will still take a long time, but it should make it a little easier.

Once you've finished your task and earned the "A Card in the Hand" achievement, you can head to the Gold Saucer and take the quest titled "The Adventurer with All the Cards" from the Gold Saucer Attendant (X: 5.0, Y: 6.7). Then, Godbert Manderville himself will gift you the Magicked Card mount for a job well done.