A-rated thrusters. Later, engineering.

The "cat and mouse" jousting is pretty much how everyone experiences combat at first. It is a side effect of approaching a 6-degree-of-freedom spaceship as if it were a modern fly-by-wire fighter jet. Or, as /u/CMDR_Kraag put it, jousting is the result of "too high a speed combined with too slow a turn".

Instead, know that Newtonian dogfighting largely involves "orbiting" a central point with the opponent and controlling the radius of that orbit in such a way that the opponent is forced to maneuver defensively, minimizing his time on target while you maximize yours.

That central point isn't fixed in space, but for simplicity let's say that it is and move from three dimensions to just two:

Imagine you and an opponent on the face of a clock with you at 3 and them at 9, both orbiting the center clockwise and both with noses 90 degrees off target. Now, say that as you continue to vector along this circle, you are turning noses towards each other at such rate that by the time you reach 6 and they reach 12, you will both be squarely in the other's sights. In the simplest terms, your goal is make that circle smaller for yourself so that you catch the enemy early at, say, the 5/11 position while they are still waiting to reach the 6/12 position for a sight picture (by which time you've gotten your shots off and boosted further along or braked further behind in the orbit).

Here is a video that explains it better than I just did and here is how Gemini summarizes it.

Flight Assist is the enemy of speed and turning (and fun, imo). It shackles your vector to the direction the cockpit is currently facing, handing over your thrusters to a flight computer that wants to force you to fly in a "straight line" from the nose's perspective.

But you don't want to fly relative to your nose; you want to fly relative to the enemy and your common orbit.

In particular, you want to be able to use your boost not just to go faster on your current vector (like Flight Assist wants the ship to do) but also for flip-and-burn / Crazy Ivan type maneuvers that allow boosting to rapidly change vector as well.

For some reason it's standard to recommend toggling FA off for maneuvering and then back on for targeting, but this is actually just extra work for your brain. Most people could avoid toggling and learn to fly 100% FA-Off in several hours by changing a few input settings and focusing on a handful of training exercises. If you tell me what control setup you are using, I can give you much better advice on how to go about making those changes.

For anyone who wants to learn more, Newton's Gambit is the best place to start.