How can I go about obtaining the activation enthalpy of a reaction using the Arrhenius Equation experimentally?
Experiments at various temperatures will allow you to calculate $E_a$. This is clear from a putting the Arrhenius equation in a different form:

$$\ln{k}=\ln{A} + \frac{-E_a}{RT}$$

A plot (or linear regression) of $\ln{k}$ vs $\frac{1}{T}$ yields $\frac{-E_a}{R}$ as the slope. The same experiment with a catalyst will show the activation energy is decreased.