ORIGINAL: ChuckW
Due to the airfoil of the trainer wing, it will tend to rise as you increase in speed. The good part of that is that it means the wing has a lot of lift which is good for a trainer.
If it were me, I'd find a throttle setting where I am comfortable flying and trim the plane to fly level there.
If you slow down or speed up for short periods, compensate with the elevator to keep it level. If you decide to speed up or slow down for much longer periods of time then maybe trim it for level flight at the new setting.
And, this is called trimming for level flight no matter when you do it. But, manually holding the elevator at that setting isn't the solution.
So, yes, you do trim for level flight after a throttle change. Get used to it. That is part of flying. Sooner or later, you will find that after you take off, you may find that your trim is the throttle. Meaning you will adjust your throttle for level flight. Often, though a click or two of elevator will do the "fine tuning" on that trim. But, when you change the throttle for a maneuver, well, once finished, you either return to that setting or trim the elevator.
CGr.