What the route is built around
Official patch notes describe five islands and an intended path. Patch 1.0.5 removed travelling to locked islands. Each step is a quest, a story boss and a trophy hand-in, then Radar coordinates for the next stop: Lighthouse, Forest, Desert, Rocks, Volcano.
Forest: leeches before bait
Giant Piranha does not come from Beginner Boss Lure. Sweep the ground for three leeches, turn them in at the lake, then use Modified Leech. Sunfish and The Old Pike are optional mini-bosses on Beginner Boss Lure.
Desert: inspect Needlefish
Catch Needlefish on Standard Lure, give it to the tourist under the tree, use the Carrot for Pufferfish. That is the Vacation achievement and the Island 4 unlock.
Do not gamble your progression money
Rocks has casino systems. They are optional. Sell duplicates, keep Tuna for Albatross, and save for Professional / Scientific lures and the boat engine upgrades.