How does an aircraft maintain level flight while accelerating? Let's assume a plane is initially moving at constant speed, level flight with a certain angle of attack. If the plane's thrust is increased, the plane's horizontal speed increases (at one point the drag force catches up and becomes again equal to thrust. The speed becomes constant at that point). 

But while the speed is increasing, the lift would increase and would the plane start climbing. How can we keep the plane in horizontal level flight while the speed is increasing? Does increasing the thrust need to be simultaneously complemented with controlling some of the rear lift surfaces to oppose the natural lift increase due to the wing?
Simply put, in the scenario stated, to maintain level flight the appropriate amount of nose down input would be applied.  

As the op points out, the increase in speed through the air increases lift, and the AOA (angle of attack) needs to be reduced.  This is normally done with stick (or yoke) input, followed by trim.

On most aircraft the lowering of the nose would be accomplished by lowering the elevator (and/or raising the trim tab) on the rear horizontal surfaces.