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You can purchase the fishing rod from Fannys shop! There is a fishing spot up in the mountains. It's the screen with the stream (left at the fork on the mountain screen) and there should be a big fish icon on the water you can click. 

There's a known bug where sometimes the fishing rod and the guidebook will appear to be sold out even if you haven't purchased them. To fix this, hit the escape button and return to the main menu. Reloading your save will have the fishing rod and guidebook back in-stock.