From what I understand the source input will have some internal resistance. So a what is the function of an input resistor or load resistor in a voltage/current amplifier circuit model? How do you analytically determine the minimum resistance values needed in order to amplify the output signal the most?
The input resistor is not a physical resistor we added because it has a desired "function"; rather the amplifier is non-ideal, and the resistor is added to make the model of the amplifier more accurate and make it able to model the interaction between the amplifier and the source impedance.
In many cases it does not correspond to a physical resistor - e.g. the input impedance \$r_\pi\$ of a common-emitter BJT amplifier is a small-signal consequence of a BJT base drawing current when voltage is applied. We didn't physically put a resistor there when building the amplifier, but the amplifier's non-ideality is modeled effectively in this way.
Another example is a common-gate MOS amplifier - while we ideally want it to have an input impedance of zero, there is a small-signal voltage that arises at the input in response to a small-signal current, which happens to physically be \$1/g_m\$.
How do you analytically determine the minimum resistance values needed in order to amplify the output signal the most?
The best gain depends on the type of amplifier - a current amplifier has the best gain when the input impedance is minimized and source impedance is maximized. A voltage amplifier has the best gain when the input impedance is maximized and the source impedance is minimized. Some of the best voltage amplifiers can have DC impedances in the giga-ohms (e.g. using JFET or MOS inputs) to achieve this goal.
