I'm happy to hear that my comments were useful.
I assume that in your previous experiment you used a microphone tied into your computer to record the sound. To use the phonograph cartridge, you would just replace the microphone with the cartridge. Radio Shack lists a cartridge at
http://www.radioshack.com/product/index ... absetBasic. This cartridge probably would work simply substituted for your microphone. If it doesn't, you might need a "phono preamplifier." again from Radio Shack, etc.
You will have to connect the cartridge to a cable so you can plug it in. The wire and plug are also available from Radio Shack and other electronic parts suppliers, but you could also cut the cable off an abandoned set of earphones since they have the same plug.
More problematic is hooking it up if you have no experience in electronic wiring. Connecting the cable to the cartridge is usually done by connectors that clip on the cartridge and have wires already attached to them, or the connectors are designed to be easy to solder wires onto them. There are too many options to discuss them all, so rummage around, look at used-stuff stores, and see what you can find. Given what you come up with, I'll provide more details. Again, if you can find an old phonograph, you could remove the cartridge and the wiring from the turntable arm, and it might be cheaper than just buying a new cartridge.
Looking at the various wave types should be interesting. One should always start out with limited goals and just expand the project if you have time. I think I would start out by placing the cartridge in contact with the side of the wine glass (or tube) and map out the response to different pitches. If you get this well characterized, then move on to observing the wave at the fluid surface.
I'm not surprised that you saw little change in pitch with varying temperature. The speed of sound in a liquid does change with temperature, but likely changed very little over the temperature range you achieved. On the other hand, the container of liquid has its response determined by its size, weight, the rigidity of the container, and other factors. It will respond most easily at certain frequencies and not so well at others.
Many parts of your setup may vary in how they respond to different pitches. You will need to consider all these. For example, if you sing the pitch to excite the liquid vibration, can you sing exactly the same volume at each different pitch? How might you standardize it?
I recommend that you read about sound in liquids and about resonance. You might start at
http://en.wikipedia.org/wiki/Underwater_acoustics and
http://en.wikipedia.org/wiki/Acoustic_resonance.
I hope you can make progress with these comments, and again, good luck.