How to maintain straight and level flight after a climb?
Attitude + Power = Performance
Performance is your airspeed, climb/descend rate, and rate of turn.
Attitude is pitch, bank, and yaw. 
Power is RPM, if you have fixed pitch prop.

So how do you transition from climb to level flight? When you reach 2000ft, or maybe 1950, push forward on the yoke or control stick to about level pitch, check the altimeter to make sure you're at 2000ft, not climbing or descending.  Your airspeed will start going up. Maintain forward pressure, it is going to increase as the speed increases (you can put some trim in to help).  Once you reach the desired airspeed, reduce throttle to keep the airspeed constant.  Then use trim to take out the pressure on the yoke.  

Now this will give you a workout, because if you don't know exact RPM that you need to maintain desired speed, you will be chasing it and working the yoke to keep constant altitude.  So I would lower the nose after the climb, let it accelerate, then lower RPM to 2200 and keep the plane in level flight with the yoke and trim.  After a few minutes when everything is stabilized, I would note the airspeed.  Then I would reduce RPM to 2100, retrim nose up to keep level, wait a few more minutes and note the airspeed for 2100 RPM.  And so forth.  You can build a table of RPM and airspeed.  Then, next time you fly, you will just set the correct RPM for desired airspeed.