Well Skyrender's guide was correct with regard to the fishing pole. Winter finished and as soon as my pond defrosted in the spring, I took all the fish I had been storing in my fridge in Winter and threw them in the pond, which was quite a few trips back and forth. But the very next day I was rewarded by Greg the fisherman, who gave me the upgraded Fishing Pole. This really improves your ability to catch fish. I have used that Pole to catch some of the King Fish. So far I have caught the Angler Fish (late night fishing in the ocean in Winter), The Char (in the river by the Hot Spring after you have cooked all the fish dishes) and the Catfish (in the underground lake in the Winter Mine). Still remaining are the Squid (can only be caught in Summer) and the Bream (can only be caught when all other King Fish are caught). I think that to catch the Squid you also need to have shipped (by hand) at least 200 fish, which means I have another 50 or so fish to catch and ship.
The best place to fish, but the furtherest away, is the lake by the Winter Mine. Although only small and medium fish are caught there, the catch rate of fish to garbage is small. It is also useful to not litter anywhere, to improve your catch rate, and its best to forgive your littering sins by confessing to Carter the priest in the confessional on Mondays or Wednesdays between 1pm - 3pm.