After spending most of my first playthrough obsessively fishing, I figured I'd share what I've learned about the best spots for each season.
Spring: The ocean spots near Half-Moon Bay and Blue Beach are great early on. Rainbow Trout, Anchovy, and Clownfish are common catches. Upgrade past the Wooden Rod to a Chrono Rod as soon as you can for better results.
Summer: Head to Coral Sea (accessible via the Parrot Ship from Starsand Port for 30 coins). The deep sea zones have the most valuable catches. Also, Lionfish and Emperor Angelfish spawn exclusively during summer rain, so watch the weather.
Fall: River fishing picks up in autumn. Chinese Perch and Red Porgy are solid sellers. If you've unlocked the Celestwave deep sea zone, you can find Ribbon Eel there at night (8pm-2am).
Winter: Lake fishing at Cloudrest Lake has winter exclusives like Baikal Sculpin (rainy nights only, very rare). The Moonlit Forest rivers (Spiritshade River zone) also have unique fish like Ink Phoenixfish that are worth checking out.
General tip: always check the weather. Some fish only spawn in rain or snow, so rainy days are prime fishing days for rare catches.