The fancy I-V curve tracers will produce a family of curves that are necessary to display transistor gain characteristics. You don't need that for a simple diode.
It's the one I recommended, but please don't assume I like it. It won't work for your diode anyway. The peak voltage from a 12VAC xformer is about 17V peak. With that low an output voltage, you'll never see the knee of the zener curve for a 79(?) volt zener. The transformer needs a higher voltage. Above about 48VDC, things become a bit dangerous. Remember, you have but one life to give for your hobby (or profession).
You don't really need to run it on AC. All AC gives you is the ability to see what's happening with both the forward and reverse conduction curves in the same plot on the oscilloscope screen. If you use a variable voltage DC power supply to "draw" the forward and reverse curves independently, you get the same result. The circuit is really simple. The diode is in series with a resistor to limit forward current. X-axis (voltage) goes across the diode. Y-Axis (current) goes across the series resistor. If your power supply has a grounded negative lead, use a 4 trace scope and differential inputs to keep the scope ground isolated.
Another way to do this is to use the sine wave output of an audio generator. Connect a step up transformer to the audio output to get up to maybe 90 volts peak. Keep the series current low (both forward and reverse) because the transformer can't handle much DC current. You should get a usable trace on the scope across the diode.
Yep. That's the way a zener works. Just make sure that when the zener conducts, the current through the zener is low enough to not blow up the zener when it conducts. Mostly, that means use a large value series resistor.