I found where the records are listed after going back to Falls End (Holland Valley). Inside the Spread Eagle bar, to the left was a board with the fish weights that I need to beat.
Thank you @Sinatar for a source to this question:
Bull Trout: 21 lbs – Jacob’s Region, northwest of Langford Lake.
Golden Trout: 4 lbs – Jacob’s Region, west of the northern bridge entrance on Silver Lake.
Lake Trout: 37 lbs – Jacob’s Region, north of Fort Drubman in Wishbone’s Lake.
Rainbow Trout: 28 lbs – John’s Region, in the small pond southwest from Rae-Rae’s Pumpkin Farm.
Holland Valley Region (John):
Record board location: Inside the Spread Eagle bar in Fall's End.
Pallid Sturgeon: 55 lbs – South of the Lamb of God Church.
Paddlefish Sturgeon: 112 lbs – Far southwest of Rye & Sons Aviation, south of the southwestern-most gun shop in the region.
Whitetail Mountain Region (Jacob):
Record board location: In the Gun Shop in Wolf's Den.
Chinook Salmon: 27 lbs – Northwest of the F.A.N.G. Center.
Kokanee Salmon: 6 lbs – Along the southwestern shore of Cedar Lake.
Arctic Grayling Salmon: 3 lbs – High up in a pond along the southwestern border of Jacob’s Region. Check your map and look for a high pond that’s far away from civilization, near the map’s boundary.
Henbane River Region (Faith):
Record board location: Close to the Arcade inside Hope County Jail.
Smallmouth Bass: 6 lbs – Along the river on the eastern edge of the territory. The hard fishing spot is between two gun stores.
Largemouth Bass: 8 lbs – North of the smallmouth bass spot, on the river close to the gun store on the eastern edge of the territory.
Rock Bass: 3 lbs – Right in the center of Rock Bass Lake.