You will be given a fishing rod by Willy, the fisherman who has a shop on the pier. If you put some time into this mini-game you can turn it into a lucrative hobby that will earn you plenty of cash.
To start the mini-game you just equip the fishing rod and head to a body of water and cast out your line. Just hold down the 'Action' button and don't release it until your cursor hits the top of the meter. When this happens your line has been cast as far as it will go and you will have the best chance possible of catching a higher value fish. You now need to be patient and wait for a fish to bite, when one does quickly tap and hold the 'Action' button the instant the exclamation point appears and try to keep the fish inside your green window long enough for the catch bar to fill up completely. If however the catch bar depletes the fish will escape.
To find fish that tend to bite more quickly and are of higher value you need to keep an eye out for bubbling spots in the water. Your fishing level will increase the more you fish and you will be able to get better rods which can cast further and have a bigger green window which will make the fish easier to catch.
Get the fiberglass rod from Willy as soon as you reach fishing level 3 as this will allow you to add bait to your line. At this level you Will also be able to buy crab pots from Willy which you can use as bait to increase your chances of catching fish. Make sure you remember to return daily to check for catch and restock the bait.