What kind of plot will be helpful for total count group data ( data grouped by month and year )?
If you like using Seaborn to construct your plots, sns.countplot is probably the best for plotting data where you want to show the differences in value counts.

You can find the link to the documentation page here which might be helpful for you to construct it in the  particular way you want.

As an example, here is a version of a "multiple" count plot like you suggest. (directly from the Seaborn Documentation page)