Requirements for Steady Level flight at various speeds
From a purely practical standpoint, let's say you're in a Cessna 172 (a common light aircraft often used for training) and you're in economy cruise at 2400 rpm at 6,000 feet on a standard day. According to the cruise performance chart I have here, you would be using 60% power, burning 6.8 gph, and have a true airspeed of 109 kts. You're hand flying the airplane as it has no autopilot.

Now the cute gal that you've talked into taking this trip with you mentions that she's beginning to need a bathroom. You neglected to bring along a urinal (and maybe more than a urinal is needed), so you decide to speed up to the maximum cruise speed for 6,000 feet. You consult the cruise performance chart and see that you can get 75% power at 2600 rpm. That will increase your cruise speed to 120 kts at the cost of burning 8.4 gph.

You would smoothly increase the rpm to 2600. As the power came up, you would have to hold a little forward pressure on the elevator to keep from climbing. If you're used to the airplane, you might unconsciously reach down to the trim wheel and roll in a little down elevator. In any event, you will eventually want to trim out the down elevator force you're holding on the control wheel.

If you set 2600 rpm to start the speed up, you'll notice that the rpm goes a little above 2600 as your speed increases (this is a fixed pitch propeller), so you monitor and adjust the rpm as necessary to settle on 2600 rpm with elevator forces trimmed.

If you have an autopilot with an altitude hold function but that doesn't control the power, all you'll have to do is control the power.

If you have an autopilot with FADEC, just set what you want and it will do all the work from there.