There are a number of website tools that I refer to over and over when I'm developing websites, and need to diagnose issues. I'm writing this list as much for myself as for anyone else. Although I have quite a few of them saved as bookmarks in Google Chrome, I figured it's useful to have them as a list on my website, as then I can look them up from any device anywhere without being logged into my Google account, and I can also provide some notes about what each site is useful for which bookmarks aren't so good for. I'll periodically update this list as I find other tools I use regularly.
I've stuck to free tools, as it's possible to spend a small fortune on subscriptions, which while useful, and often offer more than free offerings, can also be overkill, and with free offerings, it's also possible to let clients see for themselves that what you say is correct; useful in an age of fake news and skepticism.
Network Tools
Useful to find out what name servers are being used for a domain name, and the ip address of web and mail servers. Also provides some diagnostics to check that DNS is configured correctly.
If you need to set up email hosting for clients in addition to websites, mxtoolbox is a really useful tool for checking for correct DNS settings for email, and issues such as spam blacklisting.
Website Analysis Tools
Find out what hosting provider is used for a given website, and the website's ip address. Not always 100% accurate if a website is using a service like Cloudflare (which is actually a good idea security wise), as Cloudflare masks the true hosting provider, but this in itself can be useful knowledge.
This site makes a pretty good guess at what CMS (if any) a website is powered by. It can't detect everything, but uses a number of techniques to try to figure out what is running a website. This is handy if there's the prospect of taking over a website built by someone else, so you can plan costings for maintenance and development. It's also handy if you're asked to do SEO for a website, and need to know what platform it's built on so that you know what SEO tools may be available.
Pagespeed Insights (by Google)
You'd have to be living under a rock if you're in the website development or design business and aren't aware of this free website performance analysis tool from Google. Unfortunately the developers of a lot of websites seem like maybe they have been living under rocks, because it's pretty scary how badly many websites perform when run through this tool. Personally I think any website owner should check this out themselves before they pay the final instalment on any web development.
SEO
Search Console (Google)
You need to be logged in to a Google account and to have registered your website with Google to use this, however Google Search Console provides a range of useful insights about any website you control. A limitation of Search Console is that it only tracks information provided as a result of visits to your site via Google search, not visits from other sources, which you'll need a companion product like Google Analytics for. The positive side is that apart from verifying ownership of a website, you don't need to install any tracking code on your website, so privacy concerns are less than installing analytics.
This website might look a bit unusual at first glance, but it's a really useful tool for SEO, providing a detailed range of related search terms to a given search phrase. There is a paid version with additional features, but even the free version offers detailed insights that are extremely useful.
Google Trends provides a useful tool for SEO to discover popular search terms and related searches and topics.