The mediator object is intended to do nothing itself. You should not move any logic that you already have into it, except maybe for some multiplexing/demultiplexing (when one object sends the same message to multiple other objects). The mediator is just an external interface (if it simultaneously serves as a facade), and definitely a message passing channel between pre-existing objects.
Likewise, a mediator should not be created until you are already perceiving the need for such a message passing channel. How does such a need look like? You already have a set of objects that start calling each other in increasingly complex ways. Those objects are storing references to each other; the number of such references is already getting bigger than the number of such objects themselves.
So instead of each object talking to each object (with a quadratic number of references and complicated graph of interactions) you introduce a star topology to interactions; everybody directly talks just to the mediator. It is then easier to instantiate, monitor, debug, extend, polymorphize...
Do not start introducing mediators too early or the overall complexity will grow instead of dropping.