Lets start again from the beginning.
The basics of measuring any voltage in a system is by measuring the DROP of voltage ACROSS one or many components. See Kirchoff's law as to why.
As long as the voltages aren't going beyond your scopes capability (400V is what I see most on scopes, but check your own) then you can measure any two points in the circuit and get a voltage reading. What you are reading is the amount of voltage drop across that component.
In your case with the modifier and speaker, you want to see the difference in output between the input and output of the modifier. The speaker needs to be there so you have something to measure the output across. An 8-ohm resistor will also give you a reading, but it will be different from a speaker due to inductance. Speakers are rated 8-ohms of IMPEDANCE, which is resistance and inductance together. An 8-ohm resistor is measuring resistance only, since it doesn't have any inductance it's going to act somewhat different from the speaker.
Either way, you need something in your circuit to measure across; speaker or resistor. Put your grounding ring on one side of the speaker, and the probe tip on the other. Literallly measuring across the speaker. This is how you measure anything in a circuit. Nothing has to be broken to measure voltage. Just touch the two points with the probe.
Now, since you need to be able to compare the voltages you measure, you must make sure to use the same reference point in every measurement. You can literally pick anyplace in the circuit as your reference, however GND (most often the negative rail) is usually used to be consistent. Connect your ground clip to the battery/power supply negative rail, then use the probe tip to make all your other measurements. Now you have well referenced meaningful voltages.
Again, don't confuse circuit ground with earth ground. Completely different in this context.