Sorry, but you should without question use a cache. It will only mess things up for your visitors if you use the wrong settings. W3 Total Cache without minifying javascript or CSS is safe. This issue has been brought up and this solution recommended in Newspaper’s docs and the forum many times. 🙂 It will pretty much do a page cache, expiry headers, and basic (but highly necessary) compression.
W3 Total Cache can also be combined with a service like Cloudflare, which serves your site to visitors from its servers rather than your own. It speeds up delivery of some resources and helps with your server response time. (Pagespeed reports that your server responded in a massive 3.7 seconds.) I use this setup too.
A plugin like EWWW Image Optimizer will help with your optimize images problem. Note that you’ll want to do a one-time bulk optimization of your media library, which will take a long time, but after that it should just optimize on upload.
Feel free to ask for more details on the above steps. Once you try all of them, check your scores again. They will have greatly improved.
These are the steps that require little or no technical knowledge, but uf you still need more, start reading the excellent posts by Christopher Simmons and others in the sticky Pagespeed thread.
Best of luck!