Easy-to-do experiments that clearly show outstanding properties of graphene
You may want to post this in the Physics forum, as many of the outstanding properties of graphene are expected to be physical.  See this Periodic Video: Graphene

Given this, two things to try.  First, graphene should conduct electricity, like graphite but not like carbon.  Second, graphene may dissolve in nonpolar solvents like carbon tetrachloride, similar to buckyballs, to produce colorful solutions.  See this Periodic Video: BuckyBall

Both these experiments should be checked in advance: I'm extrapolating from other experiments, but have not done these myself.