How to identify whether a molecule is planar
The concept of a plane of symmetry is easier to understand with molecules which are planar by itself.  Benzene, for example, has such a plane which may accommodate simultaneously all carbon and hydrogen atoms:

However, planarity of molecules is not a requirement to identify such planes.
Even if a construction of the cyclobutane molecule with model kit illustrates you the non-planarity of the molecule

(source)
you drawing depicts it like it were planar; because you defined drawing plane where carbon atoms alternate to be either closely above, or below this level in common.  For compound 1 illustrating 1,3-dichloro cyclobutane, however, the symmetry analysis is much affected by the configuration of the two special carbon atoms.  Either you find the two chlorine atoms on the same side of this plane (syn)

or you find them directing to different sides (anti):

The interactive training set by Symmetry@Otterbein may be a helpful resource for you.  Note that the representation of compound 2 is a simplification, and the symmetry attributed strongly depends on the underlying geometry.