I'm considering installing Google's PageSpeed module on a Debian Linux / Apache 2.4 web server to speed up a Magento CE website. Will using Google's PageSpeed module help to speed up a website using Magento?
The PageSpeed module will only speed up the "front end" of the site - for example reducing HTTP requests and minifying CSS/JS.
It will certainly improve speed somewhat, but how much depends on how fast the back end (server side code) is compared to the front. Magento is generally known for being slow in the back end.
But you can check this yourself using your browser Dev Tools (Chrome/Firefox). Look under the "Network" tab at the request for the page itself (should be the first item), hover over the coloured bar to the right, and look for "Time to First Byte". (You can also see similar analysis using WebPageTest.
A large TTFB value means that the back end code is the bottleneck. In which case you may need to look for some additional plugins such as caching ones.
Using Google PageSpeed will speed up a magento based site. The improvement will be somewhere from noticeable to amazing, depending which features you do enable in your pagespeed configuration and the speed of your DB backend.
But it will also be possible that some features of your website will break or that your server will have problems to cope with the additional work.
In the end it could be possible that the effort for testing and fine tuning the pagespeed configuration is bigger than implementing the improvements directly in your website. If you don't have enough experience implementing those improvements yourself, you shouldn't use pagespeed either, or you will be lost if you run into errors.