I differenced the time series and got this plot. I think I'm supposed to use a variance stabilizing transform because variance is increasing over time but I'm a bit confused as to which one I'm supposed to use. I understand it depends on how variance increases (linearly, quadratic, etc.) but how can you tell just from the plot? Is there some step I am missing or a different way I can stabilize this time series? I'm a beginner with these concepts so any help would be appreciated!
If the variance of the model errors is proportional to the expected value the power transforms may be appropriate When (and why) should you take the log of a distribution (of numbers)? ... your data plot doesn't suggest that remedy . If the variance of the errors changes at fixed points in time then GLS following the suggestion by TSAY might be appropriate http://docplayer.net/12080848-Outliers-level-shifts-and-variance-changes-in-time-series.html but that doesn't seem to be appropriate .
Outlier detection via INTERVENTION DETECTION can be useful insofar as unusual values inflate the error variance if not mitigated.
Why don't you post your interesting time series (not a pix but the actual value and I will put it under my microscope and try to help you further.