How is Standard Gibbs free energy measured?
Generally you measure the equilibrium constant $K$ and then convert it to $\Delta G^\circ$ (note that is change in standard free energy)

$ \Delta G^\circ = -RT \ln K $  

In the case of redox reactions, this can be done with electrodes as you mention. For other reactions you usually use some spectroscopic technique (NMR, IRC, UV, etc) that can distinguish reactants from products, but calorimetry can also be used in certain cases.

Here is a video describing one example using UV.