Best advice so far. Nice and concise.
The way the OP phrased his dilemma - and a few of the suggestions - implied what I think is a bad habit: "Flying the trim". Leveling off from a climb or a descent is actually one of the simpler skills in flying, so no need to over-complicate it. The key is the PIC needs to put the nose where he or she wants it - firmly and with authority - to get the performance desired. Accuracy first; smoothness later. Undercontrolling is as much of a problem as overcontrolling. "Climbing 700 feet unintentionally" certainly implies undercontrolling. Trim only comes into it to relieve control pressure, never to move the nose.
Yes it's a "simple" skill the way you described it (put the nose where you want it), the less simple parts are knowing where you want it, sensing the difference between that and where it is, and determining how much control pressure is needed to reduce that difference to zero.
I did sense that the OP is missing the point that while there's a specific recommended sequence for pitch and power changes when leveling off from a climb, the reaction of the airplane to changes in either elevator force/position and power is not independent. Instructors tend to provide a simplified explanation of this such as "pitch controls airspeed, power controls altitude (or climb rate)" but most airplanes don't behave that way on their own. IOW if an airplane is flying level at some speed with zero elevator force input by the pilot (i.e. in trim) and you add power without applying any force to the wheel, the airplane will change all three of pitch attitude, climb rate, and airspeed with different response times for each.
All of this comes into play when "simply" leveling off and there are multiple ways to deal with it. Fast Eddie's recommendation is a good strategy though but I think it needs further explanation:
1) It's already been posted that the reason you leave the power alone at first is because you want to use the excess power to increase airspeed. You could reduce the power immediately to what's needed for your intended airspeed in level flight but the airplane's speed increase would be so gradual you might not get to the target speed before your fuel ran out. So leave the throttle alone until you get to the speed you want. Of course this assumes you know what that speed is.
2) You definitely need to pitch the nose down some but it's not really obvious how much or how quickly this needs to be done. Changing pitch more rapidly than necessary (i.e. without "smoothness") is undesirable because it's uncomfortable for passengers and therefore something you should strive for but as Fast Eddie suggested you'll learn faster if you focus on accuracy (getting the right pitch attitude) first. That said, if you're way to abrupt you can overcontrol to the point that the pitch attitude keeps changing up and down rapidly (Pilot Induced Oscillation) and this must be avoided for all sorts of reasons.
3) The "perfect" pitch attitude during the level off is not a constant, even in smooth air. Initially you need to pitch down just enough to stop climbing but as the airplane speeds up you need to lower the nose even further because as the speed builds the wings need less angle of attack to generate enough lift to hold altitude. In addition, the elevator force on the wheel increases with speed (fairly dramatically) even if you just hold the exact same pitch attitude. This is why trimming during the level off is necessary. But during the level off while the speed is still changing (i.e. before you reduce power) you don't need to get the trim "perfect", just close enough that the residual elevator force required is easy to manage.
4) Since airspeed builds slowly your focus during the leveling off process needs to be in this order: Pitch, Altitude, Trim (coarse), Airspeed, Power. Pitch is first because if your pitch is off by as little as 1° your altitude will change fairly rapidly. Altitude is next because that's what's supposed to be constant. Correct (coarse) trim is helpful but not essential (it's possible to perform an accurate and smooth level off without touching the trim but it's a lot more work and considerably more difficult in most airplanes). Unless someone replaced your engine with something with twice the HP it will take several seconds (or even minutes at high altitudes) for the airspeed to reach your target and the only thing you need to do when that happens is reduce power so an occasional glance at the ASI is all that's needed. Finally the throttle should just sit there fully open until the target speed is reached so it get's the least attention, until the airspeed hits the target.
5) Once you've set cruise power and are holding altitude it's finally time to set the trim as close to optimal as possible. You do this by holding as light a grip on the wheel as possible while maintaining the pitch attitude that results in zero climb (the smoother the air, the lighter the grip) and sensing the
average pressure on your thumb and finger then making small changes in the trim to get that average force to zero. During this time you should be holding your target altitude as well although as long as the altitude excursions are small (less than 50 ft) and very gradual they won't affect the trim enough to matter and once you get the trim correct you can return to the exact altitude with very light elevator pressure and minimal changes in airspeed.
BTW, IMO there's nothing wrong with "pre-adjusting" the trim to get it close to the eventual setting when you are deliberately changing attitude, or power. Of course to do that you need to have a pretty good idea of how much trim change is necessary and that requires experience (the pilot's or his instructor's).
The reason to leave climb power in initially is to accelerate into the cruise regime as efficiently as possible. There is no "step", but it is possible to settle into cruise on the "backside of the power curve" and to dwell there for an entire flight. Not usually a problem until near the plane's ceiling, but it's still a good habit to get into.
A bit off topic but the true "backside of the power (required) curve" occurs between stall speed and minimum sink speed which is generally only a few knots above stall speed so unless you're flirting with the airplane's absolute ceiling (something no student should ever do) or practicing slow flight this doesn't have anything to do with leveling off.
