LEVELS 1-24 SEA SLUG QUEST. Though this quest requires 30 Firemaking it's still quicker to train this skill than to level fishing out of nothing. If you complete the Sea Slug quest it will get you quick from 1 to 24 Fishing in only a few minutes (or about an hour if you're starting from 1st Firemaking level). That being said it's the fastest method at first.
LEVELS 24-58 FLY FISHING. Once you're finished with Sea Slug you can go fly fishing in Barbarian Village or even Lumbridge. This technique requires Fly Fishing Rod and feathers. Instead of to this technique, you can catch ordinary fishes in Lumbridge near the castle. Same as before you will be dropping all catched fish.
LEVELS 58-99 BARBARIAN FISHING. On these levels that you would like to catch your first member only fish. It's suggested to visit Otto's Grotto where you can grab Leaping fish (eg. You may get there by using games necklace, teleporting to Barbarian Assault minigame and running south west from that point. Here is the most efficient and best method to max your fishing level in the shortest period of time possible.
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