What variables allow one to empirically and scientifically quantify trends for learning curves? Specifically, I am trying to quantify trends in learning for certain mediums of audio-visual communication, and what I've gathered so far suggests that there are 4 distinguishable types, being linear, logarithmic (or possibly asymptotic), exponential and ogive. How can I go from peoples qualitative observations to an actual mathematical function to show these patterns for a specific medium? Would one measure...success rate over time and that's it? Or what?
Short answer
In psychophysical tests, often %correct rates are determined. Hence, training effects are often measured by determining correct rates. The ultimate outcome measures can be wildly variable, as they are dependent on the physical characteristics of the stimulus (visual, auditory, tactile, gustatory etc).

Background
Learning curves can be measured by measuring the performance on a certain task.

From what I understand of your question you are:


  ...trying to quantify trends in learning for certain mediums of audio-visual communication...  


and you are looking for


  What one measure...success rate over time and that's it? Or what?


Taking a personal vantage point here, I have measured learning effects using various auditory, tactile and visual psychophysical tests (though not a combination of them like an audio-visual test as you are planning). I will provide a few tests I have done so far to look at training effects and I will provide some basic background information on psychophysics. Please following the links if you wish to learn more on specific subjects. I have measured the following, among others:


Speech understanding by measuring the speech-recognition threshold (SRT) in noise using the Dutch Matrix test (Houben &amp; Dreschler, 2015). The SRT basically shows you the signal-to-noise ratio where speech understanding is 50% correct. In other words, it shows how much noise a listener can handle to still understand 50% of the words in the sentences heard. I've performed this test for 12 times over four several sessions and within as well as between-session learning effects were observed, and also within-run training effects (unpublished observations); 
Vibro-tactile detection threshold. Basically we asked the subject to answer in a yes/no task if they felt a stimulus and the outcome measure was that stimulus level where the correct rate was 50%. There was no learning effect observed within and between sessions. A within-run training effect was observed, which may have been due to procedural training effects (unpublished observations);
Tactile spatial acuity using (2-point discrimination); here a person was asked to answer whether one or two stimuli were felt and then, again, a percent-correct rate (here: 62.5%) score was determined ultimately expressed as that distance where correct rate was 62.5%. No training effects other than procedural learning were observed (Stronks et al, 2017);
A vibrotactile intensity-difference (JND) task, where the subject was asked to indicate whether they could feel the difference in intensity of two stimuli. Again the correct rates were measured and expressed as that intensity where the %correct scores equaled a certain threshold (Stronks et al, 2017).
Visual acuity was measured with a grating task - again percent correct is measured, but there the outcome is visual acuity, namely an angle of resolution where the %correct rate exceeds a certain threshold. There was procedural learning observed (unpublished observations). 


Note that most of the above tasks were alternative forced choice (AFC) tasks, where the threshold (%correct) is dependent on the number of choices.

References
- Houben &amp; Dreschler, Trends Hear (2015); 11(19): 1-10
- Stronks et al, Artif Organs (2017); in press