For this term: $\lambda x.(f (g x))$, what are the free and bound variables?
I'm confused as to how to expand this so it will be easier to see. If I expand this will it be $\lambda x. \lambda f. \lambda g$?
The free variables in the term you give are $f$ and $g$, $x$ is bound by the $\lambda$-binder.
I am also confused what you mean by "expanding"? But I guess you want to abstract away those free variables. If that is true, then you will get $\lambda x . \lambda f . \lambda g . f\ (g\ x)$ (or with the parameters permuted in another way.)