This guide will give you a simple instruction on how to fish, you will read about the possible locations, the pros and cons about those locations, where to buy fishing rods and where to sell fish.
Buying fishing rods can be done over at the pub in the docks area.
Go inside and speak with one of the NPC's in there to make your purchase, if you do not have any money on you an ATM can be found inside the pub. To successfully make the purchase simply click the fishing rod using your LMB (Left mouse button).
To start fishing you will have to find a source of water big enough to be able to hold fish. There are a few locations that can be used for fishing, there is the docks, the beach and Hicktown. To get started simply put your fishing rods on the side of the water and make sure the hooks are in the water, eventually a fish will take a bite or trash will get stuck on the hook, you can take these off by pressing E on the item.
Docks is the most simple place for newer people to go fishing, but beware, you are able to get mugged here for your belongings.
Though the docks does offer a very close sell point, you can go into the pub and sell your fish and also immediately deposit your cash on your bank using the ATM inside.
Beach is also a good place for fishing and you are unable to get mugged here.
You can also easily sell your fish here by going to the Fredy's.
Or you could park a van with a big trunk at the top of the stairs and load fish into it.
Hicktown is more hidden place, but just like the docks, you are able to get mugged here for your belongings.
Though if you buy a Hicktown trailer, which is almost a must if you go fishing here. You have the ability to really easily store your fish and belongings inside of the storage chest inside of your trailer.
After using the Hicktown trailer's storage you still need to sell the fish after, you can do this by going to the Suburbs Storage. You can take fish out of the storage into your inventory.
Then you can sell the fish to the Fredy's next to the Suburbs Storage.
Selling your fish is really easy, all you have to do is find a NPC vendor that sells items, it doesn't matter which NPC you use, simply interact with them by pressing E while looking directly at them, and then clicking the fish thats in your inventory. Left-click sells 1 fish at a click. Right-click sells 5 fish (or less if you dont have 5) at a click.
Thanks to Auris for helping out with information and screenshots.