Force causes both rotation and translation
In general, a force applied to a body will cause both a rotation and a translation. In the special case of a force applied in a direction through the center of mass, the body will only translate (no rotation). In the special case of a pair of equal magnitude forces applied in opposite, parallel directions displaced laterally from each other there is no translation, only rotation. With very few exceptions, airplane control surface motions result in both rotation and translation.
In the case of a tail mounted elevator, a trailing edge up motion causes downward translation and nose-up pitch rotation. For most airplane configurations the initial response of the CG (and the main gear) will be downward while the cockpit will initially rise. There will be a point along the forward fuselage where the translation and rotation are matched and opposite such that the initial reaction at that point to an elevator input does not involve any vertical acceleration. This body station (given the name "initial center of rotation" of "ICR" will experience only rotation at the initial response to an elevator position change. The ride at the ICR will be quite smooth - no surprise that this point is usually found somewhere within the First Class section!
Hope this helps.