After you catch all the "normal" fish, you'll be able to start catching King Fish. Once you've caught all the King Fish, you'll be able to brag about having caught all of the King Fish.
There will always be fish. You can complete your fishing list (all the different varieties of fish) or you can get the largest fish of each variety(size will be in gold numbers). These are personal goals though, I'm not sure that you even get a TA-DA from the Harvest Goddess.
