You can fish at various places along the map by checking out docs that have swirling water at the end.

Any time you fish, it will cost you one hour of in-game time, regardless of how long it takes you to actually catch a fish. You can only catch one fish per hour. Generally, you will catch the same type of fish per spot. I.e, Blue-fin Tuna in Spring at the beach and Rainbow Trout in Spring at the left dock in Fisherman's Spring

FISHING INSTRUCTIONS-
TLDR: Use the arrow keys to move your rod to the place you want to cast from on the map, use the space key twice (once to activate your cast and a second time to select the distance), the hook has to land on the fish as it passes by your rod to catch it.

Detailed instructions:
1. Use your arrow keys to set the fishing rod at the point on the map that you want to "cast" your hook from
2. Use the space bar to set the distance of the cast to do this:
Hit space to activate the small arrow that will go back & forth along the top of the fishing map. The distance from the left is how far you'll send your cast. Closer to the left bar, means closer to the end of your rod. > You have to hit the space bar again to make the arrow stop moving on the distance you want to cast.
3. You have to time this selection well so that the hook has time to cast and then land on the fish as it passes your rod to catch it.