I usually perform the following tests before (or mostly: after) publishing a new site:
CSS validation
HTML validation
Several performance tests (Gtmetrix, Lighthouse, Yellowlab etc.). One single test does not provide enough information.
A browsercheck does not hurt (Lambdatest).
Technical SEO tests (Ahrefs)
Always a good idea to check the JS console for anny warnings or errors.
Most services provide a decent amount of information for free. You don’t get all the good stuff, but it is usually sufficient.