Unlock Gold Saucer in Uldah, some guy will have a quest "It could happen to you" or so.
Starting TT is in the Gold Saucer... the "quest" there is a little tour of Gold Saucer also leading you to the Triple Triad section. In this section talk to the Triad Master he will explain some basics and give you your first cards.
NPCs to challenge will have a certain sign until you firsttime beat them then you need to remember them if you want to rechallenge for some reasons.

Cards are obtained on different ways:

1. Trials and Dungeons can drop them upon completition

2. Every NPC has 2-3 cards he randomly "drops" if you beat him some of them need hours "farming"


3. You can buy cards at the guy next to the Triad Master "Triad Trader" you can get some cards directly and the bronze, silver and gold sets give you 1 out of a collection upon use... pure rng (I always get Ultima from gold)

4. Every second week there is a Tournament if you greach certain amount of points you get besides MGP Platinum Packs (amount depends on performance) and they work like gold silver and bronze just have higher chance you get something rarer. Also Tournament winner gets a certain card.

Think I summoned it up

Have fun and dont become addict