Each of the Region Hubs has a chart of fishing records- you'll find them at Spread Eagle in Falls' End, by the gun shop in the Wolf's Den, and by the arcade machine in Hope County Jail.
You'll need to beat or match the existing records. Four are county-wide, nine are region-specific. The game won't actually alert you when the record is surpassed- you have to check back in at the bulletin boards. A blue ribbon means you've beat the record.
Before doing any serious fishing, you'll definitely want to complete The Admiral mission, and get the Wonderboy rod. Then it's just a matter of going to where the fish are.
County Wide Records
Golden Trout: Find the magazine of Gold Trout fishing locations in the Woodson Pig Farm in John's region (you'll visit it during the Good Samaritan mission).
One Hard Golden Trout location is just east of the Tanami residence (location of the Shipwreck Prepper Stash), in Silver Lake.
Rainbow Trout: Find the magazine of Rainbow Trout fishing locations in the actual Visitor Center part of the Whitetail Park Visitor Center.
One Hard location is well southwest of Drubman Marina, off Silver Lake in Faith's region.
Lake Trout: There's a Hard spot for them northeast of Linero Building Supply.
Bull Trout: There is a Hard spot between the Stone Ridge Chalet and Dansky's Cabin, in Jacob Territory.
Paddlefish Sturgeon: Find the magazine in Linero Building supplies, atop the watertower.
One Hard location is well southeast of Fall's End, by a branch in the Henbane.
Pallid Sturgeon: The fishing magazine for these is in the Taft Lookout Tower.
One Hard location is the riverbend northwest of the Seed Ranch, in John's region.
Rock Bass: The Rock Bass Magazine in the Breakthrough Camp.
One Hard location is off the island in the middle of Rock Bass Lake (makes sense).
Smallmouth Bass: Find the smallmouth fishing magazine at the Nelson Residence in Faith's Region.
One Hard location is just southwest of Gethsemane Greenhouse, on the east edge of Faith's region.
Largemouth Bass: A fishing magazine with largemouth bass locations is at the Loresca residence, west of Drubman Marina.
One Hard location is just slightly northwest of the Whistling Beaver Brewery.
Kokanee Salmon: There's a Hard spot for them northwest of the Grand View Hotel in Jacob's Territory.
Chinook Salmon: There is a Chinook Salmon fishing magazine inside the Prepper Stash at Dansky Cabin.
There is a Hard location for them southwest of Osprey Cabin, in Jacob's patch
Arctic Grayling There's a magazine with Arctic Grayling locations in the locked room of the Elk Jaw Lodge.
There's a Hard spot for them on the north shore of Snowshoe Lake.
Complete all fishing challenges to unlock the Old Betsy Fishing Rod.