Generally speaking there are two main "methods" to the fishing mini-game that people tend to employ.
The first is the mash method, as it sounds you mash left click really fast in bursts to keep the hook bar in line with the fish. This method is useful for catching the slower moving and lazier fish that tend to just move to one spot and hover up and down slightly.
The second method is the hold method where you actually hold down the left mouse button for a slightly longer period of time to keep the hook bar in line with where the fish is and potentially where it is going to be. You want to use this particular method for the more energetic fish that move up and down more frequently, simply hold the left mouse button down to catch up with the fish, release and then hold again as the hook bar starts to drop. The reason this is more useful for faster fish is because you want the momentum of moving up or down (from either holding the left mouse button or completely releasing it) to carry through as the fish moves.
For example, if you hook a quick fish and it immediately starts moving up to the top of the bar you will want to hold the left mouse button down to quickly catch up to it and start getting the catch bar up. Hovering over it, you will want to move up and down from tip to tip of your hook bar ready for it to move again, so when it starts dropping fast you're able to just release the left mouse button and let the momentum of your bar catch it quickly. The trick then is to get used to slowing it down as it falls by holding the left mouse button as it nears the fish in it's new spot and once again centering.
It's really really easy once you get to grips with it and there will be some ridiculously fast and crazy fish that both requires you to use the holding method and the mash method in conjunction with one another.
As expected the Tackle you use can give you an edge but the two only decent options when struggling with actually catching a fish are the Lead Bobber and the Cork Bobber. The Lead Bobber will give your hook bar some weight so it won't bounce like a maniac when it hits the bottom of the fishing bar. Very very useful when dealing with the fast fish as they can go from the very bottom of the bar to the top within seconds. The Cork bobber will simply increase the size of your hook bar which will give you more room for error when trying to center on a fish.
Pupcakes mentioned using the Barbed Hook which, while it does has its uses, I definitely wouldn't recommend on fast fish. The hook bar will try to stick to where the fish is currently and when the fish is moving so fast, you'll effecitvely lose control of where you want the hook bar to be. I would however recommend it greatly for slower fish, especially if you're just trying to catch anything as the bar does not need as much management.
Almost all the other Tackle only help with either getting a fish to bite at all (if for some reason you can't get any bites) or help with grabbing the red treasure boxes that show up from time to time. The only exception is the Trap Bobber which will cause the catch bar to reduce a bit more slowly when your hook bar isn't in line with the fish. I don't personally think it's worth it as the reduction is very minimal and when dealing with quick fish, if you lose control you will likely lose the fish regardless of whether you have the Trap Bobber or not.
If you have any questions feel free to ask!
Affism: Also try to memorize how the fish moves in the bar when you suspect you have a rare one so it would be easier next time.
This is also very useful to keep in mind. Each fish has a pattern that they will tend to stick to which can sometimes mean you know what you're going to catch before you've caught it. Memorizing whether the fish will go up or down at a given point will be very helpful in knowing when to start dropping or ascending. The faster fish will punish you for simply trying to react.