After you finish the tutorial you have a hoe, watering can, net and fishing pole. The tutorial really forces you to plant and water trees and crops. One tutorial quest is to catch a bug and they give you hints. The only clue on fishing is a sign by the North bridge that says use fish food. Fish food is expensive! I don't recommend it until you have gotten at least 15 of the 32 species.
First off we need to spot a fish. You probably saw several cruising around when your tutorial errands took you by the shore. The first rule of fishing is:
Don't cast your line in the water when there are no fish in sight. You cannot expect the fish to swim to you from far away even if you use fish food.
The best way to "hunt" fish is to use the magnifying glass in the lower left hand corner of the screen.


Even though I am standing by Polly in front of Town Hall I can see with this view two fish swimming near the channel and shore. I can use my mouse to look over the whole island and it's surrounding waters for fish.
It is better to go after one or two fish that are close rather than four that are way across the map.
By the time you get there they could have gone out of reach.



Here I am at the shore. There is a fish out there so I get the pole out of my inventory. This is now how the screen looks. The thing that took me longest to realize was that you can move the bobber position before the cast. That makes all the difference. If I cast (click on the bobber) at the current position I will not get the fishes attention and it will not strike.



The fish was swimming in a circular pattern (they often do) so I position the bobber (drag and drop) in it's path and then clicked on it. The fish is headed for it and strikes. You can see that I am offered to buy fish food, but when the fish is that close you don't need it (unless you need something very specific that has a proven particular fishfood enhancment).



Now the fish is on the line and a new icon pops up. A bobber moving back and forth across a green target area. If I click it now, I may not land the fish (it might get away).



This is the sweet spot. Click on the upper bobber icon(not moving) when the lower moving bobber is in the center of the green area.



Wow look what I caught, that baby is worth some gems!

Your ability to fish is influenced by where you position yourself. Obviously if you never get you feet wet (don't walk out in the water) you can't cast as far. You have the best chances, the bigger your casting area is.




This looks like a good place, it is on one side. On the other, obsticles get in the way.





Another question is: How close does my bobber have to be cast to the fish to make it strike?
Without fish food; pretty close, one fish length in front to actually hitting the fish works best.
With fish food;
about this far or maybe just a little farther if the fish is already swimming that way.

Here is an example of the one that got away. The fish was circling and you can see my egg fish food floating near the bobber. The fish never struck, I left it there for ten minutes. The fish circled, but never a bite. It was just too far away!
Another thing that is easy to learn the hard way is that if you take your bobber out of the water the fish food comes out with it. Don't expect to make a second cast after the fish food is applied. You can cast again but the food will be gone.

You can make money from fishing, but not near as much as from growing chili peppers or peaches.
You lose money on most of the fish you catch with bait. (the bait costs more than you can sell the fish for).
Fishing is for the fun of it, to build up the exhibit at the museum and to get the achievements.
The first step in getting the achievements then is to build the museum. In the long run, the most effiecient route is rebuilding the West Bridge first. Plant crops and grow trees there before building the North Bridge and the Museum. I am a real life collector (fossil tree xylem) so I couldn't wait, I built the museum first!
After you re-build the bridge and build the museum donate fish specimens to Harrold. You can go passed him once inside the museum into the display rooms. There you can see all your donations and view specimen cards (info) on both the ones you have and the ones you are missing.
You get the first achievement when you have donated 15 different species.
You get the second achievement after donating all 32 different species.
I think the exhibit rooms add quite a lot to the content of the game.
Be sure to visit the museum before you go to Polly (to sell your duplicate catch). Some species have similar names. Don't sell a specimen you need for the collection by mistake.

This is a copy and paste of a forum post by the developer:

Stolen Couch Games [developer] Jul 13 @ 1:41am
I can give you a bit of a spoiler:

- Basic Fishfood: Increasing size up to 10% and chance for all fish.
- Eggy Fishfood: Increased size up to 20% and chance to catch: ZEBRA_MORAY_EEL
- Flaky Fishfood: Increase chances to catch: RED_JUVENILE_YELLOWTAIL_WRASSE, RED_SLATE_PENCIL_SEA_URCHIN, RED_WALKING_CRAB
- Arctic Fishfood: Increases chances to catch: ZEBRA_MORAY_EEL, YELLOW_TRUMPET_FISH, STRIPE_SNAPPER, RAINBOW_TROUT
- Flaky Fishfood: ZEBRA_MORAY_EEL
- Soy Fishfood: RED_WALKING_CRAB
- Squid Food: BLUE_FLYING_SQUID
- Big Fishfood: Size increase up to 100%! Also bigger chance to catch: ZEBRA_MORAY_EEL, YELLOW_TRUMPET_FISH, STRIPE_SNAPPER, RAINBOW_TROUT
- Wheat Fishfood: PURPLE_QUEEN
- Worms Fishfood: YELLOW_PUFFERFISH, SOCKEYE_SALMON, STRIPE_SNAPPER, RAINBOW_TROUT

Have fun!

Out of 137 fishing successes 65 were fishfood free and on 72 I used fishfood.
Out of 65 no food successes 13 were garbage (20%).
Out of 72 fishfood sucsesses 2 were garbage (3%).
Out of 14 basic fishfood successes 2 were garbage (14%).
Out of 58 fishfood (except, no basic) sucessess 0% were garbage.
Out of 14 basic fishfood successes 3 were red fin tuna, 2 were pink tail tigerfish, 7 were one each assorted and 2 were garbage.
The 10 flaky fishfood sucsesses netted 2 pink tail tigerfish, 2 walking crab, 2 slate pencil urchin, 2 coris wrasse, 1 blue needlefish, 1 longtail bannerfish, .
5 wheat fishfood outings netted 4 purple queen fish and a 2 barred goat fish. This agrees with devs post.
5 worm fishfood outings netted 2 rainbow trout, a blue needlefish, a harliquin sweet lips, and a blue stripe snapper.
10 artic fishfood outings brought in 3 blue snapper, 2 trumpet fish, a pinktail tigerfish, a picasso tigerfish, 2 rainbow trout and a zebra eel.
5 soy fishfood ventures brought in a striped boxfish, a walking crab, a red fin tuna, a trumpet fish, and a lontail banner fish.
3 egg fishfood ventures brought in an eel, a coris wrasse, and a blue moon fish.
10 big fishfood ventues brought in 4 rainbow trout 40,47,50,51; 4 zebra eels 123,154,170,289; a trumpet fish 73, and a blue snapper 24. (numbers are length rounded off.)
That research set me back some gems! It also agrees with the devs forum post.
Thirty out of thirty two species were caught.Two fish were not caught in the sampling. They were spotted eagle ray and bullet head parrot fish.
Five species were caught only once. They were blue moon fish, bi-color goat fish, grey moon fish, saddle back grouper, and red lion fish.
Seven species were caught without any fishfood.
The 2 barred goat fish were caught with either basic or wheat fishfood.
The four purple queen fish were only caught while using wheat fishfood.

Now get out there and don't let the big one get away!