How can I improve my fishing ability?

Fishing in Animal Crossing can be tricky sometimes, even for seasoned players. This is because it has gone through some subtle iterations, so the best "method" of fishing may not be the same across different AC games, and is often hotly debated online.

Informal consensus across previous games in the series is that it was preferable to rely on your other senses -- for example, listening to the sound of the bobber or feeling for the rumble of a bite in games that had it. It is easy to flinch or get distracted by the many things happening in the game's scene, and restricting some of that distraction may prove helpful. New Leaf doesn't support any sort of "rumble pack," so as far as this method is concerned you would be left to depend on sound. You can try this by closing your eyes, and holding down on A as soon as you hear the appropriate "bite" sound.

A more reliable technique is to note the "5 bite limit" of fish in Animal Crossing. They won't ever bite more than five times. If you're waiting for the fifth bite, then it may be better to have your eyes open to hold down A just as soon as the fish touches the bobber.

After you progress far enough in the game, you will have the opportunity to acquire a "silver" or "gold" fishing rod. These upgrades will make catching fish easier because they will bite down longer. Hopefully by the time you are eligible for those upgrades, your fishing prowess will make them unnecessary! Otherwise...

Silver Fishing Rod:

Catch at least 30 different types of fish and upgrade the museum to have a second floor. Then you can purchase this from the museum's second floor.

Gold Fishing Rod:

Catch all types of fish in the game, and then speak to the fishing tournament host.

Well, it's combination of two factors: primarily your personal fishing practice, and secondarily, your fishing rod.

For the primary reason, learn to tune out the other sounds and sights of the game, and focus squarely on the rod and pond. Learn to listen for your rod's bobs, as well as the "bites" (the time to hold down A). It is frustrating at first (especially with the more difficult and pricy fish) but, you'll learn to gain the hang of it with experience. This is the most reliable one out of the two reasons.

For the secondary reason, you may upgrade your fishing rod twice over the course of the game. You start out with a Fishing Rod, and can upgrade it to a Silver Fishing Rod, and later, the Gold Fishing Rod. The benefits of doing so are that the upgraded rods cause the fish to bite down for longer periods of time, providing ampler opportunities for pressing down "A" on time.

The requirements for the Silver Fishing Rod are that: you need to catch at least 30 species of fish and build the second story of the museum, where you may purchase this rod. The requirements for the Gold Fishing Rod are that: you need catch all types of fish present in the game, and talk to fishing tournament host.

Well, as you have seen, your best bet is to raise your personal experience through repeated practice. I hope you found my answer helpful. Happy fishing!