The solution above is great on how to get Old Betsy Fishing Rod but that is only 1 of the 4. The 4 Rods you need to unlock for the achievement are:
Basic Fishing Rod can be bought for $300 from the shop
Natural Fishing Rod can be bought for $800 from the shop
Wonderboy Fishing Rod is unlocked by completing Skylar missions
The Admiral and
Gone Fishin'Old Betsy Fishing Rod is unlocked by catching fish that beat the County Record
Holland Valley Region - Fall's End, inside Spread Eagle Bar in the back left.
Pallid Sturgeon - 55 lbs
Paddlefish Sturgeon - 112 lbs
Herbane River Region - Hope County Jail, door to the left of the shop entrance
Smallmouth Bass - 6 lbs
Largemouth Bass - 8 lbs
Rock Bass - 3 lbs
Whitetail Mountain Region - In Wolfs Den behind the shop counter
Chinook Salmon - 27 lbs
Kokanee Salmon - 6 lbs
Arctic Grayline Salmon - 3 lbs
All Regions (Hope County) - Left side on any of the 3 boards
Bull Trout - 21 lbs
Golden Trout - 4 lbs
Lake Trout - 37 lbs
Rainbow Trout - 28 lbs
Here is a Video Guide to help with Old Betsy thanks to
Maka91:
You have to fish in the spots marked as "hard" on the map ie: Golden Trout (Hard) located directly south of Rye and Sons Aviation. Also I found it really helps if you change your bobber by first holding down
then while holding
press
and while the Fishing Rod is highlighted press
to switch your bobber to the one that is the same as the fish you are trying to catch. These bobbers are:
Basic Mayfly
Hybrid Sturgeon
Salmon Egg
Bass Popper
Trout Bugger