There's a total of 34 fish species. The in-game encyclopedia will show you a summary of each one after you've caught it, including its preferred season and time of day. If you are playing the Special Edition of Light of Hope, the encyclopedia will also list the largest size and smallest size of each fish type you've discovered.

The below information is what I've caught in the game. Some fish might appear in areas other than what I've confirmed.

Fish Name	Season	Time of Day	Location	Bait	Sell Price
Rainbow Trout	Summer Fall	6:00 am - 4:00 pm	Beach Pier Farm (Stream) Lighthouse Pier Mountain (Stream)	(None)	Normal: 900 G Bastian: 1350 G
Mosquitofish	All Year	All Times	Everywhere	(None)	Normal: 200 G Bastian: No buy
Perch	All Year	All Times	Beach Central/Eastern (Stream) Farm (Pond) Mountain (Pond) Mountain (Stream)	(None)	Normal: 100 G Bastian: 150 G
Yellow Perch	All Year	All Times	Farm (Stream and Pond) Mountain (Stream and Pond)	(None)	Normal: 200 G Bastian: 300 G
Goby	Summer Fall	All Times	Beach Central/Eastern (Stream) Beach Pier Farm (Stream) Lighthouse Pier Mountain (Stream)	Regular	Normal: 600 G Bastian: 900 G
Barbatus	Spring Summer Fall	6:00 am - 4:00 pm	Farm (Pond) Mountain (Pond)	(None)	Normal: 2000 G Bastian: No buy
Bullhead	Fall Winter	4:00 pm - 5:00 am	Farm (Stream) Lighthouse Pier Mountain (Stream)	Regular	Normal: 1700 G Bastian: 2550 G
Arowana	Fall	11:00 am - 7:00 pm	Farm (Pond and Stream) Mountain (Pond)	Good	Normal: 2000 G Bastian: No buy
Bluegill	Spring Summer Fall	11:00 am - 5:00 am	Beach Central/Eastern (Stream) Farm (Pond and Stream) Mountain (Pond and Stream)	(None)	Normal: 600 G Bastian: 900 G
Crucian Carp	All Year	6:00 am - 11:00 am 4:00 pm - 5:00 am	Beach Central/Eastern (Pond and Stream) Farm (Pond and Stream) Mountain (Pond and Stream)	(None)	Normal:500 G Bastian: No buy
Alligator Gar	Summer	6:00 am - 4:00 pm	Mountain (Pond)	Tropical	Normal: 11,300 G Bastian: No buy
Akame	Fall Summer	7:00 pm - 5:00 am	Mountain (Stream, at the base of the waterfall)	Delectable	Normal: 8500 G Bastian: No buy
Green Arowana	Fall	11:00 am - 7:00 pm	Farm (Pond and Stream) Mountain (Pond and Stream)	Excellent	Normal:4700 G Bastian: Nob buy
Banjar Arowana	Fall	11:00 am - 7:00 pm	Farm (Pond and Stream)	Good	Normal: 11,300 G Bastian: No buy
Golden Arowana	Fall	11:00 am - 7:00 pm	Mountain (Pond)	Excellent	Normal: 14,500 G Bastian: No buy
Super Red Arowana	Fall	11:00 am - 7:00 pm	Beach Eastern (Pond)	Ragnarok	Normal: 18,500 G Bastian: No buy
The only spot I could find Super Red Arowana is the pond on the east edge of the Beach: Eastern area, between the gap in the map-edge fence. That one pond also has Banjar Arowana and Green Arowana.
White Crucian Carp	All Year	6:00 am - 11:00 am 4:00 pm - 5:00 am	Beach Central/Eastern (Stream) Farm (Pond and Stream) Mountain (Pond and Stream)	Regular	Normal: 1300 G Bastian: No buy
Goldfish	Spring Fall	6:00 am - 11:00 am 4:00 pm - 5:00 am	Farm (Pond, Raining) Mountain (Pond)	Good	Normal: 4200 G Bastian: No buy
Calico Goldfish	Summer Winter	6:00 am - 11:00 am 4:00 pm - 5:00 am	Mountain (Pond)	Good	Normal: 7600 G Bastian: No buy
Spotted Gar	Summer Fall	6:00 am - 7:00 pm	Farm (Pond)	Tropical	Normal: 4000 G Bastian: No buy
Black Bass	Spring Fall	6:00 am - 4:00 pm	Beach Central/Eastern (Stream) Farm (Pond and Stream) Mountain (Stream)	(None)	Normal: 1000 G Bastian: No buy
Tilapia	Spring Summer	6:00 am - 7:00 pm	Beach Pier Farm (Stream) Lighthouse Pier Mountain (Stream)	(None)	Normal: 400 G Bastian: 600 G
Redbelly Tilapia	Spring Summer	6:00 am - 7:00 pm	Beach Pier Lighthouse Pier Mountain (Stream)	Regular	Normal: 1000 G Bastian: 1500 G
Arapaima	Spring Summer	11:00 am - 7:00 pm	Farm (Stream) Mountain (Stream)	Tropical	Normal: 12,700 G Bastian: No buy
Piranha	All Year	All Times	Farm (Pond and Stream) Mountain (Stream)	(None)	Normal: 600 G Bastian: 900 G
Piraya Piranha	All Year	All Times	Farm (Pond and Stream) Mountain (Pond and Stream)	Excellent	Normal: 1600 G Bastian: 2400 G
Electric Eel	Summer	7:00 pm - 5:00 am	Beach Central (Ponds)	Tropical	Normal: 18,000 G Bastian: No buy
Tuna	Winter	6:00 am - 4:00 pm	Beach Pier Lighthouse Pier	Good	Normal: 6000 G Bastian: 9000 G
Sardine	All Year	6:00 am - 4:00 pm	Beach Pier Lighthouse Pier	(None)	Normal: 300 G Bastian: 450 G
Cod	Summer Winter	6:00 am - 4:00 pm	Beach Pier Lighthouse Pier	Good	Normal: 1300 G Bastian: 1950 G
Sea Bream	Spring	6:00 am - 4:00 pm	Lighthouse Pier	Delectable	Normal: 2000 G Bastian: 3000 G
Striped Beakfish	Fall	6:00 am - 4:00 pm	Lighthouse Pier	Good	Normal: 2000 G Bastian: 3000 G
For the Striped Beakfish, stand in the middle of the pier and cast north (towards the lighthouse) or south (towards the sea).
Righteye Flounder	Spring Winter	6:00 am - 11:00 am 4:00 pm - 7:00 pm	Beach Pier	Good	Normal: 2500 G Bastian: 3750 G
Olive Flounder	Fall Winter	6:00 am - 11:00 am 4:00 pm - 7:00 pm	Beach Pier	Regular	Normal: 2500 G Bastian: 3750 G