I just did something like this:
I hung out by the river until grabbing all the fish for that season at least once then I moved to the beach and lake next. Rinse and repeat until getting all the common fish every season. And when the Midnight Market was on, went every day to grab multiple copies of the rare fish. That way I got all the fish requirements for the fishing archivement.
If you are focusing on only getting the bundles, it should go something like this:
Spring
Sunfish : River , until 7pm
Catfish : River and secret woods when raining
Shad : River when raining, after 9am
Largemouth Bass : Lake, until 7pm
Carp: Lake, Secret Woods, Sewer
Bullhead: Lake
Sardine: Ocean, until 7pm
Eel: Ocean, when raining, after 4pm
Chub: Lake and river (Demetrius bundle)
Summer
Sunfish : River, until 7pm
Shad : River when raining, after 9am
Largemouth Bass : Lake, until 7pm
Carp: Lake, Secret Woods, Sewer
Bullhead: Lake
Sturgeon: Lake, until 7pm
Tuna: Ocean, until 7pm
Red Snapper : Ocean when raining, until 7pm
Tilapia: Ocean, until 2pm
Bream: River after 6pm
Pufferfish: Ocean, after 12pm to 4pm, sunny
Chub: Lake and river (Demetrius bundle)
Fall
Catfish : River and secret woods when raining
Shad : River when raining, after 9am
Tiger Trout : River, until 7pm
Largemouth Bass : Lake, until 7pm
Carp: Lake, Secret Woods, Sewer
Bullhead: Lake
Sturgeon: Lake, until 7pm
Sardine: Ocean, until 7pm
Red Snapper : Ocean when raining, until 7pm
Tilapia: Ocean, until 2pm
Walleye: River, Lake, Forest pond when raining, after 12pm
Bream: River after 6pm
Eel: Ocean, when raining, after 4pm
Chub: Lake and river (Demetrius bundle)
Winter
Tiger Trout : River, until 7pm
Largemouth Bass : Lake, until 7pm
Carp: Secret Woods, Sewer
Bullhead: Lake
Sardine: Ocean, until 7pm
Tuna: Ocean, until 7pm
Walleye: River, Lake, Forest pond when raining (totem), after 12pm
Bream: River after 6pm
Chub: Lake and river (Demetrius bundle)
Crab Pot
Ocean: Lobster, Crab, Shrimp, Cockle, Mussel, Oyster, Clam
Freshwater: Crayfish, Snail, Periwinkle
Beach foraging: Cockle, Mussel, Oyster, Clam, Sea Urchin (this later for Emily bundle, at the Tide Pools or during green ocean)
Monster drops: Crab (Rock and lava crabs in the Mines)
Other
Ghostfish: The Mines floors 20 and 60, drop by ghosts
Sandfish: Desert until 8pm
Woodskip: Secret woods
Void Salmon: Gold or Iridum quality, Witch's swamp (Missing bundle)
Nautilus Shell: Beach foraging during winter (Demetrius bundle)
Maki Roll: made using seaweed, rice and any fish. Can be bought randomly at Krobus' shop on Saturday or Stardrop Saloon. You can buy the recipe at the Saloon or watching Queen of Sauce on 21 Summer year 1. (Gus bundle)
Rain-only fish:
Catfish : River; Spring, Fall
Shad: River; Spring, Summer, Fall; after 9am
Eel: Ocean; Spring, Fall ; after 4pm
Red Snapper: Ocean; Summer, Fall; until 7pm
Tilapia: Ocean; Summer; until 2pm
Walleye: River, Lake; Fall, Winter; after 12pm
Among the lake ones, only the sturgeon isn't available all year round so these ones you can do whenever you have some time to kill.
Edit: I edited his post to add everything related to fish.