How can I formulate a rate equation using a solid catalyst? When formulating a rate a equation two experiments must take place: one in which the reactant concentrations are varied, and one in which the catalyst concentrations are varied. 

If I am given a sample of a solid catalyst, such as $MnO_2$, how can I change its concentration? Because I can't dilute a solid and increase the volume, which is how I'd do it for a catalyst in solution.
You will perform the reaction inside a container or solution of some particular volume. This means that you can change the catalyst concentration inside that volume by adding more or less catalyst in different experiments. So you don't increase the volume of the solid (i.e. change its density), you change the mass of solid inside the reaction volume.