Fishing is very easy once you get the hang of it and it just becomes more simple as you level up the skill. The larger and more rare the fish the more exp you get for fishing.
The first place you can fish is in Galdin Quay. It is to the south of where you start. If you follow the main story line you will be taken to this location for a level 5 quest. Just go out to the beach and then find the docks. There is a merchant that will sell you everything you need in order to fish.
Fish will respawn in the pools given time. You can probably just go sleep and come back and they will have returned. When you see the reel icon in the center of your screen that means the line is getting tension and it will start causing the line HP to degrade. If the line breaks you lose the bait as well. Otherwise the bait is able to be reused. When looking at the minimap before you cast you can see little dots. Those are the fish. The closer you land your lure to the fish the better chance you get for getting their attention.
Once you have their attention use press the right trigger once to pull the lure toward you. Let it sit for 1-2 seconds and then pull it again. You are basically having to entice the fish to get the lure. When you first cast if your lure is close enough sometimes the fish will bite it instantly. So try to line up target perfectly before you cast.
Fish seem to like several different types of lure. I've managed to catch every first on the minimap at Galdin Quay with only 1 lure. Some fish prefer certain lures over others but it seems that they will eventually bite on more than just one type of lure.