We've got a Codeignter website - NOT WordPress - that needs speeding up to get a decent score on PageSpeed Insights and Pingdom (and in real life browsing)
The site is template based - there are two main page templates that need fixing.
There are also a number of CSS files that could usefully be combined
Pingdom is also reporting that some file types need caching - presumably some extra lines in htaccess.
There are also some "behind the scenes" mySQL requests - a custom written mySQL query (this should be already done but needs to be checked) and various other queries that I think could be slowing things down.
It's also reporting a long server response time - the site is on a Plesk based VPS with 3Gb of RAM and SSD disks so should be a fast response time but is being reported as about a 6 second response time.
Please include your experience on speeding up Plesk based sites and preferably include a site that you've speeded up that wasn't WordPress based.