Earlier quoted context omitted.
Every time I look at a website its shocking how no one even does basic image optimisation using PS on the images. This is such a trivial optimisation it makes you wonder about the level of competency of the average web development company.
Is there an alternative that doesn't require a monthly subscription?
For JPEGs: use chroma downsampling, and don't go above 80% quality unless absolutely necessary. If your software has a preview function, use it. Encode with MozJPEG if you can.
For PNGs: compress the hell out of them. There's several tools that can compress them further.