How to experimentally measure damping in RLC circuits?
With Fc the LC resonant frequency, you can :


Excite the circuit with a square wave of much lower frequency that Fc. 


Each edge of the square wave will produce exponentially decaying ringing, ie you will see the transient response, and you will know if it is damped or ringing (underdamped).

Then, you can measure the Q factor by looking at how the ringing decvays. If you solve the circuit's differential equation, you will get an exponentially decaying sine. Measure the decay between one peak and another peak a bit later, also measure the time (or number of periods) and you can compute Q.


Bode plot


You can use a network analyzer, or in LF you can simply use a soundcard and some freeware audio program to plot the frequency response of your circuit when excited by a source of a known impedance. Then, read this question :

Resonant Frequency from Bode plot

Note: in both cases, make sure you take the output impedance of the signal generator into account.