Find Oleg in your farm. Click on him and select "go fishing". Once the mini-game opens, make sure that your mouse pointer is below the water when trying to reel in fish and is not over any of the icons on the left such as the bait or reel icons, nor on the bar above the water which shows the strength of your line and is not used to reel fish in. These only show what equipment you have brought with you or other stats about your equipment and will not allow you to reel in fish. You should know when you have been successful as the mouse will change from a hand to a reel shape when you click to reel fish in.

Now keep clicking left clicking while moving the mouse circularly to reel the fish in.

 

When using your own bait (to catch a specific kind of fish), first you have to buy it and put it in your tackle box. Then enter the mini-game. Click on the bait you would like to use on the left hand side of the screen. A new button will appear with a green arrow symbol, simply click this to use one piece of bait and check the quest screen to see if it has registered for you.