How can I configure the plot labels so that when I scale the plot (either manually or producing a high-resolution rasterized version of it), they scale too? Basically, I want the labels to take up a constant fraction of the overall plot no matter what size the plot is
To show the plot or just about any part of the notebook in larger size, you can use
Magnify. You can also set the magnification option of the notebook (lower right corner).
To rasterize with higher resolution, set the resolution, not the image size.
Rasterize[..., ImageResolution -> a 72]
will rasterize at a size
a times larger than the
ImageSize, if you like to count in pixels, or at
a 72 dpi if you like to think in terms of printing.
You can specify the font size in
Scaled units to allow text to scale when you resize the image, but I do not recommend this. Instead, I suggest to prepare figures to a certain print size as I described here. This will make it easy to match up the size of the text in your figure with the surrounding text if the figure will be inserted into a document. It also allows you to rasterize at the correct print resolution (typically 600 dpi for line art).
I wrote about these ideas in: