The easiest way to find the resonant frequencies is to place the object next to a speaker and also place a microphone attached to an oscilloscope next to the object. Have the speaker play a tone at a given volume, and then without changing the volume slowly change the pitch (or frequency). If you watch the oscilloscope you will notice that at certain frequencies the amplitude of the wave, which is proportional to the volume of the sound being picked up by the microphone, will be greater than at surrounding frequencies. These are the resonant frequencies, and are detectable as the sound energy absorbed by the object is re-emitted more efficiently at these pitches.
Note that you can perform the same procedure, albeit less precisely, in a low-tech way: try holding a large bowl, or coffee can, or some other object that you are hoping to make resonate, in front of your face. Slowly sing a tone with increasing pitch. If there is a resonant frequency in the audible range, you should be able to hear the tone emitted back to your ears at that pitch. Or if you have a piano available to you, try singing into the piano and you'll see the strings vibrate when you sing their resonant frequencies.
Answered by: Rob Landolfi, Science Teacher, Washington, DC