Fishing is Noctis' skill and can be done at any fishing spot you find in the wild. When you get to one, press X/A to cast your line, then move it around until you happen to hook a fish. Some fish will only react to certain lures, so you'll either need to find them or purchase them at various shops. You'll know a fish is interested if a red line appears at the top of your screen.
Once the fish is hooked, you can reel it in using R2/RT, but there are a few things that you should know before doing this. You want to always tilt your left stick in the direction the fish is swimming, as this will reduce the chances of tension on the line. Also, if the fish is struggling, as indicated by the controller vibrating or the fish splashing, try to avoid reeling. If the tension prompt appears, stop reeling and tilt the stick in the direction the fish is swimming to make it disappear.