Fishing is a chill gathering skill. You catch fish to cook for food. It is useful for all account types, especially Ironmen who need a steady supply of healing items in  OSRS gold.
Early Levels (1 to 20)
No quests, just fish:
From levels 1 to 5, catch Shrimp with a small net south of Lumbridge. From levels 5 to 20, switch to a fishing rod and bait. Catch Sardines and Herring at the same spot on OSRS GP.
Better way using quests:
Do Sea Slug first. It requires level 30 Firemaking and gets you to level 24. Then do Fishing Contest. It requires level 10 Fishing and gets you to level 26. These two quests skip the slowest part of early training.

Mid Levels (20 to 58)
Use fly fishing. You need a fly fishing rod and feathers. Fish north of Ralio in Varlamore or at Shilo Village. You will catch Trout at level 20 and Salmon at level 30. Stay here until level 50 or 58. Experience per hour starts around 10,000 to 12,000 and reaches 30,000 to 32,000 at level 50.

Useful Minigames
Fishing Trawler (level 15 and above)
This is a short five-minute fishing trip. Your reward is the Angler outfit, which gives you 2.5 percent more Fishing experience. Get this early because it helps forever.
Tempoross (level 35 and above)
This is a fishing boss minigame. You need a harpoon, rope, hammer, and buckets of water. The gameplay loop is simple: fish, then either cook the fish for more rewards or keep them raw for more experience, then shoot the fish at the boss using crates. The best reward is the Fishing Barrel, which holds an extra 28 fish. Experience per hour ranges from 35,000 to 85,000 depending on your level and gear.

AFK Methods (Low Attention)
These are great for watching shows or doing other things.
Barbarian Fishing is available at level 58 and above. You need a barbarian rod and feathers. Drop your fish when your inventory is full. There is no profit. Experience per hour is 50,000 to 65,000.
Karambwans are available at level 65 and above. You need to complete the Tai Bwo Wannai Trio quest. This method is very AFK. Use a Fishing Barrel to double your inventory. Experience per hour is about 30,000. Profit is 200,000 to 300,000 gold per hour.
Monkfish are available at level 62 and above. You need to complete the Swan Song quest. This method is easy but low profit. Experience per hour is about 40,000.

Profitable Methods (Good Money)
Infernal Eels require level 80 Fishing and a Fire Cape. This method is very AFK. Use a hammer on the eels when your inventory is full. Experience per hour is about 30,000. Profit is about 220,000 gold per hour.
Dark Crabs require level 85 Fishing. This method is in the Wilderness, so it is risky but offers the best money. Use the nearby NPC to note your fish. Experience per hour is about 40,000. Profit is about 400,000 gold per hour.
Sacred Eels require level 87 Fishing. This method gives Zulrah scales instead of raw fish. Your profit depends on your Cooking level. Experience per hour is about 25,000. Profit is about 300,000 gold per hour.