I would like some tips on how to build a circuit that will let me verify the impedances of these basic components. I had tried just measuring the output voltage from the component, but that didn't seem to work. I have an Analog Discover 2, it's a small oscilloscope/function generator. It can only measure voltage. Also the function generator can go up to 10 MHz, at that frequency will I see in inductive of capacitive effects in a typical carbon core resistor?
A common method is the bridge method using a variable resistance with a signal generator and obtain 1/2V voltage then assume the ESR is neglible and compute or look up the measured pot resistance on an LCR nomograph. You can vary the frequency from 100 Hz to 1MHz also to choose area of interest for its application.
Make copies of this to markup.
To avoid the inductance of a pot your resistor reference must have very short leads. So the above method fails. To measure the inductance of a resistor , a better method is use a large non inductive plastic cap with no leads and resonate it with a series resistor to find the resonant frequency. Calibration with a scope probe with no ground leads is also required to avoid these measurement errors too. ( remove tip and clip and use coaxial barrel for measurements)
This is not an exhaustive how to measurement guide.
Although LCR meters use a constant current source at a fixed frequency and measure voltage as impedance and phase shift to get Q, DF, ESR etc. they may exceed your budget, but are the best tool for the job.
If you can afford $100~$150 for a decent one, they are worth it.