The Fishing class is obtainable in Limsa Lominsa. All that's required to fish is a rod and some appropriate bait for your fishing location. The description of the bait will say where it should be used. The types are Ocean, Freshwater (lakes, rivers etc), Dunefishing (sand) and Skyfishing (Yep, that's a thing!). Each fish caught will use up a bait, so I suggest you stock up on them if you're planning to grind this.
Attach the bait, cast your line and wait for the line to tug. Your controller will vibrate, which makes it very afk'able. If you're watching the screen, you'll see the end of the rod being pulled downwards. It's easier to see with the Cast Light ability.