How to calculate the amplitude increase by resonance? If you know the natural frequency of an object, how would one calculate how much the amplitude increases because of resonance? I would imagine it would depend on the material?
 A: In general you need to know both the strength of the driving force, its frequency, as well as the loss coefficient of the equations. 
A driven, damped simple harmonic oscillator in one dimension satisfies:
$$\frac{d^{2}z}{dt^{2}} + \gamma \frac{dz}{dt} + \omega_{0}^{2} z = \frac{F_{0}}{m}e^{j\omega t} $$
Note that this describes the response of a simple system to a specific frequency of driving force.
It makes intuitive sense that the solution would have a frequency equal to that of the driving force. However, it might have a phase factor. 
Thus, we assume a solution of the form $ z = A e^{j(\omega t - \phi)} $, and then we can solve for A and see what the response will be as a function of the driving force's frequency.
Plugging in, we find: $$ (-\omega^{2} + \omega_{0}^{2} + j\gamma \omega)A e^{j\omega t} = \frac{F_{0}}{m} e^{j\omega t} e^{j\phi} $$
Since both sides are complex, we get two equations in two unknowns, $ A $ and $ \phi $. 
After some algebra, we get that $$ A(\omega) = \frac{F_{0}/m}{((\omega_{0}^{2} - \omega^{2})^{2} + (\gamma \omega)^{2})^{1/2}} $$.
That's the result you want.
You asked about its dependence on the medium. The medium will determine the loss coefficient $ \gamma $. As $ \gamma $ gets larger, the denominator increases and the amplitude of the oscillation decreases. 
This derivation is in Vibrations and Waves by A.P. French, along with a great deal more of information.
