This isn’t so much a “job” as it is a recreational activity. However, if you really like to fish, you can fish from your boat, in streams, or from docks and sell your catch to make money.
If you want your own boat, you’ll need to buy one from 8239 (blue sail boat icon) for most boats or if you want more fancy yacht style boats, talk to the guy at postal 8238.
You can buy your fishing gear from the guy at postal 1080. You can also rent a quad from here if you wanna go hunting or drive somewhere off road to fish.
Angler/Offshore rod can be used from boats, piers, ledges, etc. Fly rods require that you be wading in at least some water of a stream, river, beach, etc. Use fly fishing lures for fly rods, of course, and buy tackle boxes for the Angler/Offshore rod. You will get a random assortment of lures from the tackle box to use with that rod, and the tackle box will be removed from your inventory.
Use the rod you want from your inventory and follow the prompts.
If you want to sell your catch, you can do so by talking to the guy at postal 3012.