Here are a few that are available for free. You will see recommendations on what you can improve:
https://www.screamingfrog.co.uk/seo-spider/ (SEO and Crawl tool, free for up to 500 pages)
https://moz.com/ (mix of free and paid SEO tools)
Here is a good tool for monitoring page loading speeds https://www.host-tracker.com/InstantCheck/Create?type=Waterfall This is a very good tool, given that Google recently did not correctly display this figure. There are other indicators that can be checked for free for a month. Among others there are tools for receiving notifications of problems on the site. For search engines I can not advise something specific, in this case it is better to audit the site.