Elevator trim sets the hands free angle of attack the airplane will seek on its own, for a given weight, center of gravity, and power configuration. In the absence of an angle-of-attack indicator, we use airspeed as a proxy for angle of attack, since they are directly related.
You generally let the trim do most of the work of holding a desirable pitch attitude for you, except for short term maneuvering inputs. So you should be in the habit of trimming out control pressures to let the airplane do the work of holding a speed/attitude for any case where you need the airplane to fly like that for more than a short time, say, 15 to 30 seconds. That way, if your attention is diverted and you relax your vigilance momentarily, no worries, as the plane will hold the speed/attitude you want largely on its own except for minor deviations due to turbulence or some such. You should think of the trim wheel as a hands-free speed control dial.
So if you're at, say, 70 kt and you are holding forward stick pressure to keep the nose from rising and the airplane slowing down, you trim nose down until the nose stays where you want it hands off. This leaves you free to pay attention to other things while you fly. Don't over think it. If you are holding forward pressure, trim ND, if back pressure trim NU until the nose attitude and speed stay where you want them without your input.
With tractor engine power planes with the engine in the nose, there is an added complication you must account for. Changes in power affect trim, either because the thrust line is above or below the center of mass and causes pitching moments, or because the increase in propeller wash over the tail increases the tail's effectiveness. Generally the change in tail effectiveness dominates, and you find that an increase in power changes trim in the nose-up direction (because the prop is "blowing" on the tail harder).
So, trim speed will often drop when you add power. Experiment yourself in the airplane to see the effect: Set the plane up in level flight, trimmed hands off to hold 80 kt, then add power to climb, and leave the stick alone and just watch what the airplane does. It will hunt up and down for a while, seeking, then settling down at it's original trim state, but now in a climb because the power was increased. But you'll notice the trimmed speed is no longer 80kt. It will be some knots lower, say, 75 kt. Reverse the effect for power reductions. A reduction in power for a tractor engine will result in higher trimmed speed when everything settles down. It's as if a passenger climbed from the front seat to the back seat when you add power, and vise versa.
As you get on top of this phenomenon, you will anticipate it and learn to add a slight amount of nose down trim when adding power to climb (assuming you want to maintain the same speed) in order to keep the airplane from seeking a slower trim speed than you originally set. This is important safety wise in situations like go-arounds where you are firewalling the throttle with the plane trimmed to its final approach speed. If you do nothing, and relaxed on the controls due to some other distraction, the power increase will reduce the trim speed and the pitch up will be more than that required to convert the extra power to climb. You might find yourself approaching stall in a high power climb situation if you weren't paying attention to pitch.
