Are you using Wordpress? If so, there are many good plugins for most things you need to do to speed up your website. I actually been doing this myself today, went from mobile 13 to 57 and desktop 93 with just installing and config a few plugins. I will continue tomorrow and see if i can get mobile speed even higher.
Just search in google and you will find plenty of guides how to do it.
Some plugins I installed today (all free):
Clearfy - Many features like optimize js, css and html, turn off google fonts, image optimization, turn off many default wordpress features you never use...etc
Enable Gzip Compression - Not sure if good or not but google suggest you do it so.
W3 Total Cache - I used this to manually add the js and css links that PageSpeed Insight showed that slowed down my website.
WP-Optimize - Cleans up your database.
There are many more plugins you can use to speed up your website and I will try some more but I don't like to install too many plugins but if they can speed up my site I'm all for it. Just do it slowly and check your website working properly after each change you do as well as run a new speed test and see if it did any difference.