There are many different SEO analysis tools out there that can test your site and give you a report of how you are doing in regards to SEO. While a lot of what you are given is helpful, some of the information that may seem "concerning" as reported by these tests may not actually be something to worry about. Here is some information on what to pay attention to.
The nice thing that PageSpeed Insights does is that it gives you both a mobile and a desktop analysis of your site. Keep in mind, mobile is almost always a lower score than Desktop. Mobile devices have a much different process they go through to load your site, which is oftentimes a longer loading process. The less heavy the content (i.e. big images, embed codes, etc.) the faster your site will potentially load, and the better it will score on these tools.
**It is okay to sacrifice loading time according to the analysis tools if your business depends on it! Photography heavy, content-filled sites will take a little bit longer to load in than a site that only has text on it. Just figure out what excess you aren't needing on there, and these tools can help show you that.
You can click through the different Opportunities section to learn more about your performance score. Some of this information is helpful like "Use Video Formats for Animated Content". This tells me that taking the gifs off of this example site and instead use a video, it would improve speed. It even includes a load time saved estimate.
Another suggestion is made on this example to "Eliminate Render-Blocking Resources." This is one of the examples of things not to worry about, as there isn't a way to alter any of that as it is something your site needs to run (Showit-site required elements).
Pingdom is another great tool that allows you to analyze your site from locations all across the globe.
After checking your site you will be presented with a similar list of suggestions to improve page performance. You can tab through the suggestions, but keep in mind they are just suggestions and not all of the low scoring items may be able to be altered. The analysis told the user to "Add Expires Headers" and "Compress Components with Gzip" giving both of those sections really low scores. These are not things that can be altered based on the way that your site is built, the tool is just doing its best to make suggestions on what it analyzes.
One of the greatest benefits of this tool is that it has a filter section that you can browse through to find out different things like what files take the longest to load, which files load in at what times, and other helpful data. At the bottom of Pingdom under File Requests, it's helpful to Sort By > Load Time and uncheck the Rising selection.
You will then be presented with an accurate list of how long it takes elements to load on your site, and what size they are. You can then go down the list and select the items to see a visual of what is slowing down the page load time. After finding those elements, it can be beneficial to compress them to shrink their file sizes, and then reupload them. Even though Showit does compress files when uploading, a smaller file pre-compression can actually help out.
At the end of the day these tools are really helpful and can be utilized to improve your site, just keep in mind that some of these suggestions may be out of your control. If you focus on the parts that you do have control over, primarily your site content, you will be making big improvements to your site!