which tools can I use to monitor & improve my sites performance in search engines?

Here are a few that are available for free. You will see recommendations on what you can improve:

https://developers.google.com/speed/pagespeed/insights/

https://gtmetrix.com/

https://www.pingdom.com/

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.

For monitoring page load speed: Google Page Speed, GTMetrix, Pingdom. For SEO and crawling you can use Screaming Frog SEO up to 500 pages are available for free. Also, Moz has a mix of free and paid tools you can use for SEO.