Minion question from new player

[Enkidoh]

All of the above. Some can be bought from certain shops, some are rewards from quests (including main scenario quests), some are found in dungeons, some are crafted or even grown in house gardens, a couple are drops from certain high level FATEs, some are bonuses that came with real-world merchandise, some are won by unlocking certain achievements, some are veteran rewards for subscribing to the game for a certain length of time, and some are bought on the Mogstation for real money.

This page has a list of all the known (and currently available) minions and where to find them.

Just be aware that a couple of minions are now completley unattainable, as they were either only available during short-term special events (like the wind-up Shantotto), or were gained as a voucher that came with merchandise that has since expired (like the wind-up Dalamud).